GIFT   OF 

MICHAEL  REESE 


b 

-V 


WORKS   OF 
PROF.   WALTER   L.   WEBB 

PUBLISHED    BY 

JOHN  WILEY  &  SONS. 


Railroad  Construction.— Theory  and  Practice. 

A  Text-book  for  the  Use  of  Students  in  Colleges 
and  Technical  Schools.  x-|~456  pages  and  18 
plates.  8vo.  Cloth,  $4.00. 


Problems  in  the  Use  and  Adjustment   of  Engin- 
eering Instruments. 

Forms  for  Field-notes  ;  General  Instructions  for 
Extended  Students'  Surveys.  i6mo.  Morocco, 
$1.00. 


RAILROAD   CONSTRUCTION 


THEORY  AND  PRACTICE. 


A  TEXT-BOOK  FOR  THE  USE  OF  STUDENTS  IN 
COLLEGES  AND  TECHNICAL  SCHOOLS. 


BY 

WALTER    LORIXG   WEBB,    C.E., 

Associate  Member-  American  Society  of  Civil  Engineers: 

Assistant  Professor  of  Civil  Engineering  in 

the  University  of  Pennsylvania; 

etc. 


FIRST  EDITION. 
FIRST  THOUSAND. 


NEW    YORK: 

JOHN   WTLEY   &  S01STS. 
LONDON  :    CHAPMAN  &  HALL,   LIMITED. 
1900. 


Copyright,  1899, 

BY 
WALTER  LORING   WEBB. 


ROBERT  DRUMMOND,   PRINTER,   NEW  YORK. 


PEEFACE. 


THE  preparation  of  this  book  was  ^egun  several  years  ago, 
when  much  of  the  subject-matter  treated  was  not  to  be  found  in 
print,  or  was  scattered  through  many  books  and  pamphlets,  and 
was  hence  unavailable  for  student  use.  Portions  of  the  book 
have  already  been  printed  by  the  mimeograph  process  or  have 
been  used  as  lecture-notes,  and  hence  have  been  subjected  to 
the  refining  process  of  classroom  use. 

The  author  would  call  special  attention  to  the  following 
features : 

a.  Transition  curves ;  the  multiform-compound-curve  method 
is  used,  which  has  been  followed  by  many  railroads  in  this  country ; 
the  particular  curves  here  developed  have  the  great  advantage  of 
being  exceedingly  simple,  and  although  the  method  is  not  theo- 
retically exact,  it  is  demonstrable  that  the  differences  are  so 
small  that  they  may  safely  be  neglected. 

I.  A  system  of  earthwork  computations  by  means  of  a  slide- 
rule  (which  accompanies  the  volume)  which  enables  one  to  com- 
pute readily  the  volume  of  the  most  complicated  earthwork 
forms  with  an  accuracy  only  limited  by  the  precision  of  the 
cross-sectioning. 

c.  The  4t  mass  curve  "  in  earthwork;  the  theory  and  use  of 
this  very  valuable  process. 

d.  Tables  I,  II,    III,  and  IV  have  been  computed  ab  novo. 
Tables  I  and  II  were  checked  (after  computation)  with  other 
tables,  which  are  generally  considered  as  standard,  and  all  dis- 
crepancies were  further  examined.      They  are  believed  to  be 
perfect. 

iii 


83040 


iv  PREFACE. 

e.  Tables  Y,  YI,  YII,  and  IX  Lave  been  borrowed,  by  per- 
mission, from  "  Ludlow's  Mathematical  Tables."  It  is  believed 
that  five- place  tables  give  as  accurate  results  as  actual  field  prac- 
tice requires.  Tables  YIII  and  X  have  been  compiled  to  con- 
form with  Ludlow's  system. 

The  author  wishes  to  acknowledge  his  indebtedness  to  Mr. 
Chas.  A.  Sims,  civil  engineer  and  railroaJ  contractor,  for  read- 
ing and  revising  the  portions  relating  to  the  cost  of  earthwork. 

Since  the  book  is  written  primarily  for  students  of  railroad 
engineering  in  technical  institutions,  the  author  has  assumed  the 
usual  previous  preparation  in  algebra,  geometry,  and  trigo- 
nometry. 

WALTER  LORING  WEBB. 
UNIVERSITY  OF  PENNSYLVANIA, 
PHILADELPHIA, 
Jan.  1,  1900. 


TABLE  OF  CONTENTS. 


CHAPTER   I. 
RAILROAD  SURVEYS. 

PAGE 

RECOXNOISSANCE 1 

1.  Character  of  a  reconnoissance  survey.  2.  Selection  of  a  general 
route.  3.  Valley  route.  4.  Cross-country  route.  5.  Mountain  route. 
6.  Existing  maps.  7.  Determination  of  relative  elevations.  8.  Hori- 
zontal measurements,  bearings,  etc.  9.  Importance  of  a  good 
reconnoissance. 

PRELIMINARY  SURVEYS 8 

10.  Character  of  survey.  11.  Cross-section  method.  12.  Cross- 
sectioning.  13.  Stadia  method.  14.  "First"  and  "second  "pre- 
liminary survey. 

LOCATION  SURVEYS 13 

15.  "Paper  Location."  16.  Surveying  methods.  17.  Form  of 
Notes. 

CHAPTER  II. 
ALIGNMENT. 

SIMPLE  CURVES is 

18.  Designation  of  curves.  19.  Length  of  a  subchord.  20.  Length  of 
a  curve.  2^.  Elements  of  a  curve.  22.  Relation  between  T,  E,  and  4. 
23.  Elements  of  a  1°  curve.  24.  Exercises.  25.  Curve  location  by 
deflections.  26.  Instrumental  work.  27.  Curve  location  by  two 
transits.  28.  Curve  location  by  tangential  offsets.  29.  Curve  loca- 
tion by  middle  ordinates.  30.  Curve  location  by  offsets  from  the 
long  chord.  31.  Use  and  value  of  the  above  methods.  32.  Obstacles 
to  location.  33.  Modifications  of  location.  34.  Limitations  in  loca- 
tion. 35.  Determination  of  the  curvature  of  existing  track.  36. 
Problems. 


vi  TABLE  OF  CONTENTS. 

PAGE 

COMPOUND  CURVES 37 

37.  Nature  and  use.  38.  Mutual  relations  of  the  parts  of  a  com- 
pound curve  having  two  branches.  39.  Modifications  of  location. 
40.  Problems. 

TRANSITION  CURVES 43 

41.  Superelevation  of  the  outer  rail  on  curves.  42.  Practical  rules 
for  superelevation.  43.  Transition  from  level  to  inclined  track. 
44.  Fundamental  principle  of  transition  curves.  45.  Multiform  com- 
pound curves.  46.  Required  length  of  spiral.  47.  To  find  the  ordi- 
nates  of  a  l°-per-25-feet  spiral.  48.  To  find  the  deflections  from  any 
point  of  the  spiral.  49.  Connection  of  spiral  with  circular  curve  and 
with  tangent.  50.  Field-work.  51.  To  replace  a  simple  curve  by  a 
curve  with  spirals.  52.  Application  of  transition  curves  to  compound 
curves.  53.  To  replace  a  compound  curve  by  a  curve  with  spirals. 

VERTICAL,  CURVES 61 

54.  Necessity  for  their  use.  55.  Required  length.  56.  Form  of 
curve.  57.  Numerical  example. 

CHAPTER  III. 

EARTHWORK. 
FORM   OF   EXCAVATIONS    AND   EMBANKMENTS 64 

58.  Usual  form  of  cross-section  in  cut  and  fill.  59.  Terminal  pyra- 
mids and  wedges.  60.  Slopes.  61.  Compound  sections.  62.  Width 
of  roadbed.  63.  Form  of  subgrade.  64.  Ditches.  65.  Effect  of 
sodding  the  slopes,  etc. 

EARTHWORK  SURVEYS 72 

66.  Relation  of  actual  volume  to  the  numerical  result.  67.  Pris- 
moids.  68.  Cross-sectioning.  69.  Position  of  slope-stakes. 

COMPUTATION  OF  VOLUME 76 

70.  Prismoidal  formula.  71.  Averaging  end  areas.  72.  Middle 
areas.  73.  Two  level  ground.  74.  Level  sections.  75.  Numerical 
example,  level  sections.  76.  Equivalent  sections.  77.  Equivalent 
level  sections.  78.  Three-level  sections.  79.  Computation  of  prod- 
ucts. 80.  Five-level  sections.  81.  Irregular  sections.  82.  Volume 
of  an  irregular  p-ismoid.  83.  True  prismoidal  correction  for  ir- 
regular prismoids.  84  Numerical  example  ;  irregular  sections ; 
volume,  with  true  prismoidal  correction.  85.  Volume  of  irregular 
prismoid,  with  approximate  prismoidal  correction.  86.  Illustration 
of  value  of  approximate  rules.  87.  Cross-sectioning  irregular  sections. 
88.  Side-hill  work.  89.  Borrow-pits.  90.  Correction  for  curvature. 
91.  Eccentricity  of  the  center  of  gravity.  92.  Center  of  gravity  of 
side-hill  sections.  93.  Examples  of  curvature  correction.  94.  Accu- 


TABLE  OF  CONTENTS.  vii 

PAGE 

racy  of  earthwork  computations.      95.  Approximate  computations 
from,  profiles. 

FORMATION  OF  EMBANKMENTS Ill 

96.  Shrinkage  of  earthwork*.  97.  Allowance  for  shrinkage.  98. 
Methods  of  forming  embankments. 

COMPUTATION  OP  HAUL 116 

99.  Nature  of  subject.  100.  Mass  diagram.  101.  Properties  of 
the  mass  curve.  102.  Area  of  the  mass  curve.  103.  Value  of  the 
mass  diagram.  104.  Changing  the  grade  line.  105.  Limit  of  free 
haul. 

COST   OF   EARTHWORK 126 

106.  General  divisions  of  the  subject.  107.  Loosening.  108.  Load- 
ing. 109.  Hauling.  110.  Choice  of  method  of  haul  dependent  on 
distance.  111.  Spreading.  112.  Keeping  roadways  in  order.  113. 
Repairs,  wear,  depreciation,  and  interest  on  cost  of  plant.  114.  Super- 
intendence and  incidentals.  115.  Contractor's  profit.  116.  Limit  of 
profitable  haul. 

BLASTING , 142 

117.  Explosives.  118.  Drilling.  119.  Position  and  direction  of 
drill-holes.  120.  Amount  of  explosive.  121.  Tamping.  122.  Ex- 
ploding the  charge.  123.  Cost.  124.  Classification  of  excavated 
material.  125.  Specifications  for  earthwork. 

CHAPTER  IV. 

TRESTLES. 

126.  Extent  of  use.  127.  Trestles  vs.  embankments.  128.  Two 
principal  types. 

PILE  TRESTLES 155 

129.  Pile  bents.  130.  Methods  of  driving  piles.  131.  Pile-driving 
formulae.  132.  Pile-points  and  pile-shoes.  133.  Details  of  design. 
134.  Cost  of  pile  trestles. 

FRAMED  TRESTLES 162 

135.  Typical  design.  136.  Joints.  137.  Multiple-story  construc- 
tion. 138.  Span.  139.  Foundations.  140.  Longitudinal  bracing. 
141.  Lateral  bracing.  142.  Abutments. 

FLOOR  SYSTEMS 167 

143.  Stringers.  144.  Corbels.  145.  Guard-rails.  146.  Ties  on 
trestles.  147.  Superelevation  of  the  outer  rail  on  curves.  148.  Pro- 
tection from  fire.  149.  Timber.  150.  Cost  of  framed  timber 

trestles. 


Till  TABLE  OF  CONTENTS. 

PAGE 

DESIGN  OP  WOODEN  TRESTLES 174 

151.  Common  practice.  152.  Required  elements  of  strength.  153. 
Strength  of  timber.  154.  Loading.  155.  Factors  of  safety.  156. 
Design  of  stringers.  157.  Design  of  posts.  158.  Design  of  caps  and 
sills.  159.  Bracing. 

CHAPTER  V. 

TUNNELS. 

SURVEYING 185 

160.  Surface  surveys.  161.  Surveying  down  a  shaft.  162.  Under- 
ground surveys.  163.  Accuracy  of  tunnel  surveying. 

DESIGN 190 

164.  Cross-sections.  165.  Grade.  166.  Lining.  167.  Shafts. 
168.  Drains. 

CONSTRUCTION 195 

169.  Headings.  170.  Enlargement.  171.  Distinctive  features  of 
various  methods  of  construction.  172.  Ventilation  during  construc- 
tion. 173.  Excavation  for  the  portals.  174.  Tunnels  vs.  open  cuts. 
175.  Cost  of  tunneling.  I 

CHAPTER  VI. 

CULVERTS  AND  MINOR  BRIDGES. 

176.  Definition  and  object.     177.  Elements  of  the  design, 
AREA.  OF  THE  WATERWAY 203 

178.  Elements  involved.  179.  Methods  of  computation  of  area. 
180.  Empirical  formulae.  181.  Value  of  empirical  formulae.  182. 
Results  based  on  observation.  183.  Degree  of  accuracy  required. 

PIPE  CULVERTS 208 

184.  Advantages.  185.  Construction.  186.  Iron-pipe  culverts. 
187.  Tile-pipe  culverts. 

Box  CULVERTS -. 212 

188.  Wooden  box  culverts.  189.  Stone  box  culverts.  190.  Old- 
rail  culverts. 

ARCH  CULVERTS 215 

191.  Influence  of  design  on  flow.  192.  Example  of  arch-culvert 
design. 

MINOR  OPENINGS 216 

193.  Cattle-guards.  194.  Cattle-passes.  195.  Standard  stringer 
and  I-beam  bridges. 


TABLE  OF  CONTENTS.  ix 

CHAPTER  VII. 

BALLAST. 

PAGE 

196.  Purpose  and  requirements.  197.  Materials.  198.  Cross- 
sections.  199.  Methods  of  laying  ballast.  200.  Cost. 

CHAPTER  VIII. 

TIES 
AND   OTHER  FORMS   OP  RAIL   SUPPORT. 

201.  Various  methods  of  supporting  rails.  202.  Economics  of  ties. 
WOODEN  TIES 227 

203.  Choice  of  wood.  204.  Durability.  205.  Dimensions.  206. 
Spacing.  207.  Specifications.  208.  Regulations  for  laying  and 
renewing  ties.  ^209.  Cost  of  ties. 

PRESERVATIVE  PROCESSES  FOR  WOODEN  TIES 232 

210.  General  principle.  211.  Vulcanizing.  212.  Creosoting.  213. 
Buruettiziug.  214.  Kyanizing.  215.  Wellhouse  (or  zinc-tannin) 
process.  216.  Cost  of  treating.  217.  Economics  of  treated  ties. 

METAL  TIES 238 

218.  Extent  of  use.  219.  Durability.  220.  Form  and  dimensions 
of  metal  cross-ties.  221.  Fastenings.  222.  Cost.  223.  Bowls  or 
plates.  224.  Longitudinals. 

CHAPTER  IX. 

RAILS. 

225.  Early  form*.  226.  Present  standard  forms.  227.  Weight 
for  various  kinds  of  traffic.  228.  Effect  of  stiffness  on  traction.  229. 
Length  of  rails.  230.  Expansion  of  rails.  231.  Rules  for  allowing 
for  temperature.  232.  Chemical  composition.  233.  Testing.  234. 
Rail  wear  on  tangents.  235.  Rail  wear  on  curves.  236.  Cost  of  rails. 

CHAPTER  X. 

RAIL-FASTENINGS. 
RAIL-JOINTS 255 

237.  Theoretical  requirements  for  a  perfect  joint.     238.  Efficiency 
of  the  ordinary  angle  bar.     239.  Effect  of  rail-gap  at  joints.     240. 
Supported,  suspended,  and  bridge  joints.    241.  Failures  of  rail  joints. 
242.  Standard  angle-bars.     243.  Later  designs  of  rail-joints. 
TIE-PLATES 260 

244.  Advantages.  245.  Elements  of  the  design.  246.  Methods  of 
setting. 


X  TABLE  OF  CONTENTS. 

PAGE 

SPIKES 263 

247.  Requirements.  248.  Driving.  249.  Screws  and  bolts.  250. 
Wooden  spikes. 

TRACK-BOLTS  AND  NUT-LOCKS 266 

251.  Essential  requirements.  252.  Design  of  track-bolts.  253. 
Design  of  nut-locks. 

CHAPTER  XI. 

SWITCHES  AND  CROSSINGS. 

SWITCH  CONSTRUCTION 271 

254.  Essential  elements  of  a  switch.  255.  Frogs.  256.  To  find 
the  frog  number.  257.  Stub  switches,  258.  Point  switches.  259. 
Switch-stands.  260.  Tie-rods.  281.  Guard-rails. 

MATHEMATICAL  DESIGN  OF  SWITCHES 278 

262.  Design  with  circular  lead  rails.  263.  Effect  of  straight  frog- 
rails.  264.  Effect  of  straight  point-rails.  265.  Combined  effect  of 
straight  frog  rails  and  straight  point-rails.  266.  Comparison  of  the 
above  methods.  267.  Dimensions  for  a  turnout  from  the  OUTER  side 
of  a  curved  track.  268.  Dimensions  for  a  turnout  from  the  INNER 
side  of  a  curved  track.  269.  Double  turnout  from  a  straight  track. 
270.  Two  turnouts  on  the  same  side.  271.  Connecting  curve  from  a 
straight  track.  272.  Connecting  curve  from  a  curved  track  to  the 
OUTSIDE.  273.  Connecting  curve  from  a  curved  track  to  the  INSIDE. 
274.  Crossover  between  two  parallel  straight  tracks.  275.  Crossover 
between  two  parallel  curved  tracks.  276.  Practical  rules  for  switch- 
laying. 

CROSSINGS 300 

277.  Two  straight  tracks.  278.  One  straight  and  one  curved  track. 
279.  Two  curved  tracks. 


APPENDIX.    THE  ADJUSTMENTS  OF  INSTRUMENTS 303 


TABLES. 

I.  Radii  of  curves 314 

II.  Tangents  and  external  distances  to  a  1°  curve. 318 

III.  Switch  leads  and  distances 321 

IV.  Transition  curves 322 

V.  Logarithms  of  numbers 325 

VI.  Logarithmic  sines  and  tangents  of  small  angles 345 

VII.  Logarithmic  sines,  cosines,  tangents,  and  cotangents 348 

VIII.  Logarithmic  versed  sines  and  external  secants 393 

IX.  Natural  sines,  cosines,  tangents  and  cotangents 439 

X.  Natural  versed  sines  and  external  secants 444 

XI.  Useful  trigonometrical  formulae 449 

INDEX , 45! 


RAILROAD   CONSTRUCTION 


CHAPTER   I. 
RAILROAD   SURVEYS. 

THE  proper  conduct  of  railroad  surveys  presupposes  an 
adequate  knowledge  of  almost  the  whole  subject  of  railroad 
engineering,  and  particularly  of  some  of  the  complicated  ques- 
tions of  Railroad  Economics,  which  are  not  generally  studied 
except  at  the  latter  part  of  a  course  in  railroad  engineering,  if 
at  all.  This  chapter  will  therefore  be  chiefly  devoted  to 
methods  of  instrumental  work,  and  the  problem  of  choosing  a 
general  route  will  be  considered  only  as  it  is  influenced  by  the 
topography  or  by  the  application  of  those  elementary  principles 
of  Railroad  Economics  which  are  self-evident  or  which  may  be 
accepted  by  the  student  until  he  has  had  an  opportunity  of 
studying  those  principles  in  detail. 

RECONNOISSANCE     SURVEYS. 

1.  Character  of  a  reconnoissance  survey.  A  reconnoissance 
survey  is  a  very  hasty  examination  of  a  belt  of  country  to  de- 
termine which  of  all  possible  or  suggested  routes  is  the  most 
promising  and  best  worthy  of  a  more  detailed  survey.  It  is 
essentially  very  rough  and  rapid.  It  aims  to  discover  those 
salient  features  which  instantly  stamp  one  route  as  distinctly 
superior  to  another  and  so  narrow  the  choice  to  routes  which 
are  so  nearly  equal  in  value  that  a  more  detailed  survey  is  nec- 
essary to  decide  between  them. 


2  EAILEOAD   CONSTRUCTION.  §  2. 

2.  Selection   of  a  general  route.     The   general    question   of 
running  a  railroad  between  two  towns  is  usually  a  financial  rather 
than  an  engineering  question.      Financial  considerations  usually 
determine  that  a  road  must  pass  through  certain   more   or  less 
important  towns   between   its  termini.      When  a  railroad  runs 
through  a  thickly  settled  and  very  flat  country,  where,  from  a 
topographical  standpoint,  the  road  may  be  ran  by  any  desired 
route,  the  c '  right-of-way  agent"  sometimes  has  a  greater  influ- 
ence in  locating  the  road  than  the  engineer.     But  such  modifi- 
cations of  alignment,  on  account  of  business  considerations,  are 
foreign  to  the  engineer's  side  of  the  subject,  and  it  will  be  here- 
after assumed  that  topography  alone  determines  the  location  of 
the  line.      The  consideration  of  those  larger  questions  combin- 
ing finance  and   engineering  (such  as   passing  by  a  town  on  ac- 
count of  the  necessary  introduction  of  heavy  grades  in  order  to 
reach  it)  is  likewise  ignored. 

3,  Valley  route,     This  is  perhaps  the  simplest  problem.      If 
the  two  towns  to  be   connected  lie  in  the  same  valley,   it  is 
frequently  only  necessary  to  run  a  line  which  shall  have  a  nearly 
uniform  grade.      The  reconnoissance  problem  consists  largely  in 
determining  the  difference  of  elevation  of  the  two  termini  of 
this  division  and  the  approximate  horizontal  distance  so  that  the 
proper  grade  may  be  chosen.      If  there  is  a  large  river  running 
through  the  valley,  the  road  will  probably  remain   on  one  side 
or   the   other  throughout  the  whole  distance,   and  both  banks 
should  be  examined  by  the  reconnoissance  party  to  determine 
which  is  preferable.      If  the  river  may  be  easily  bridged,   both 
banks  may  be  alternately  used,  especially  when  better  alignment 
is  thereby  secured.     A  ,  river  valley  has  usually  a  steeper  slope 
in  the  upper  part  than  in  the  lower  part.      A  uniform  grade 
throughout  the  valley  will  therefore  require  that  the  road  climbs 
up  the  side  slopes  in  the  lower  part  of  the  valley.     In  case  the 
u  ruling  grade  "  *  for  the  whole  road  is  as  great  as  or  greater 

*  The  ruling  grade  may  liere  be  loosely  defined  as  the  maximum  grade 
which  is  permissible.  This  definition  is  not  strictly  true,  as  may  be  seen  later 
when  studying  Railroad  Economics,  but  it  may  here  serve  the  purpose. 


§  5.  RAILROAD  SURVEYS.  3 

than  the  steepest  natural  valley  slope,  more  freedom  may  be 
used  in  adopting  that  alignment  which  has  the  least  cost — 
regardless  of  grade.  The  natural  slope  of  large  rivers  is  almost 
invariably  so  low  that  grade  has  no  influence  in  determining  the 
choice  of  location.  When  bridging  is  necessary,  the  river 
banks  should  be  examined  for  suitable  locations  for  abutments 
and  piers.  If  the  soil  is  soft  and  treacherous  much  difficulty 
may  be  experienced  and  the  choice  of  route  may  be  largely 
determined  by  the  difficulty  of  bridging  the  river  except  at 
certain  favorable  places. 

4.  Cross-country  route.     A  cross-country  route  always  has  one 
or  more"  summits  to  be  crossed.      The  problem  becomes  more 
complex  on  account  of  the  greater  number  of  possible  solutions 
and  the  difficulty  of  properly  weighing  the   advantages  and  dis- 
advantages of  each.      The  general  aim  should  be  to  choose  the 
lowest  summits  and  the  highest  stream  crossings,  provided  that 
by  so  doing  the  grades  between  these  determining  points  shall 
be  as  low  as  possible  and  shall  not  be  greater  than  the  ruling 
grade  of  the  road.     Nearly  all  railroads  combine  cross-country 
and  valley  routes  to  some  extent.      Usually  the  steepest  natural 
slopes  are  to  be  found  on  the  cross-country  routes,  and  also  the 
greatest  difficulty  in  securing  a  low  through  grade.    An  approx- 
imate determination  of  the  ruling  grade  is  usually  made  during 
the  reconnoissance.      If  the  ruling  grade   has  been  previously 
decided  on  by  other  considerations,  the  leading  feature  of  the 
reconnoissance  survey  will  be  the  determination  of  a  general 
route  along  which  it  will  be  possible  to  survey   a  line  whose 
maximum  grade  shall  not  exceed  the  ruling  grade. 

5.  Mountain  route.     The  streams  of  a  mountainous  region 
frequently  have  a  slope  exceeding  the  desired  ruling  grade.   In 
such  cases  there  is  no  possibility  of  securing  the  desired  grade 
by  following  the  streams.      The  penetration   of  such  a  region 
may  only  be  accomplished  by  ''development" — accompanied 
perhaps   by   tunneling.      ' i  Development ' '   consists  in   deliber- 
ately increasing  the  length  of  the  road  between  two  extremes 
of  elevation  so  that  the  rate  of  grade  shall  be  as  low  as  desired. 


4  RAILROAD  CONSTRUCTION.  §  5. 

The  usual  method  of  accomplishing  this  is  to  take  advantage  of 
some  convenient  formation  of  the  ground  to  introduce  some 
lateral  deviation.  The  methods  may  be  somewhat  classified  as 
follows : 

(a)  Kunning  the  line  up  a  convenient  lateral  valley,  turning 
a  sharp  curve  and  working  back  up  the  opposite  slope.  As 
shown  in  Fig.  1,  the  considerable  rise  between  A  and  B  was 


FIG.  1. 

surmounted  by  starting  off  in  a  very  different  direction  from 
the  general  direction  of  the  road ;  then,  when  about  one-half  of 
the  desired  rise  had  been  obtained,  the  line  crossed  the  valley 
and  continued  the  climb  along  the  opposite  slope,  (b)  Switch- 
back. On  the  steep  side-hill  BCD  (Fig.  1)  a  very  considerable 
gain  in  elevation  was  -accomplished  by  the  switchback  CD. 
The  gain  in  elevation  from  B  to  D  is  very  great.  On  the 
other  hand,  the  speed  must  always  be  slow :  there  are  two  com- 

JT  t/ 

plete  stoppages  of  the  train  for  each  run ;  all  trains  must  run 
backward  from  C  to  D.  (c)  Bridge  spiral.  "When  a  valley  is 
so  narrow  at  some  point  that  a  bridge  or  viaduct  of  reasonable 
length  can  span  the  valley  at  a  considerable  elevation  above  the 
bottom  of  the  valley,  a  bridge  spiral  may  be  desirable.  In 


§6. 


RAILROAD  SURVEYS. 


Fig.  2  the  line  ascends  the  stream  valley  past  A,  crosses  the 
stream  at  .Z?,  works  back  to  the  narrow  place  at  C,  and  there 
crosses  itself,  having  gained  perhaps  100  feet  in  elevation, 
(d)  Tunnel  spiral.  This  is  the  reverse  of  the  previous  plan. 


FIG.  2. 


FIG.  3 


It  implies  a  thin  steep  ridge,  so  thin  at  some  place  that  a  tunnel 
through  it  will  not  be  excessively  long.  Switchbacks  and 
spirals  are  sometimes  necessary  in  mountainous  countries,  but 
they  should  not  be  considered  as  normal  types  of  construction. 
A  region  must  be  very  difficult  if  these  devices  cannot  be 
avoided. 

Rack  railways  and  cable  roads,  although  types  of  mountain 
railroad  construction,  will  not  be  here  considered. 

6.  Existing  maps.  The  maps  of  the  U.  S.  Geological  Survey 
are  exceedingly  valuable  as  far  as  they  have  been  completed. 
So  far  as  topographical  considerations  are  concerned,  they 
almost  dispense  with  the  necessity  for  the  reconnoissance  and 
"  first  preliminary"  surveys.  Some  of  the  State  Survey  maps 
will  give  practically  the  same  information.  County  and  town- 
ship maps  can  often  be  used  for  considerable  information  as  to  the 
relative  horizontal  position  of  governing  points,  and  even  some 


6  RAILROAD  CONSTRUCTION.  §  7. 

approximate  data  regarding  elevations  may  be  obtained  by  a 
study  of  the  streams.  Of  course  such  information  will  not  dis- 
pense with  surveys,  but  will  assist  in  so  planning  them  as  to 
obtain  the  best  information  with  the  least  work.  When  the 
relative  horizontal  positions  of  points  are  reliably  indicated  on  a 
map,  the  reconnoissance  may  be  reduced  to  the  determination 
of  the  relative  elevations  of  the  governing  points  of  the  route. 

7.  Determination  of  relative  elevations.  A  recent  description 
of  European  methods  includes  spirit-leveling  in  the  reconnois- 
sance work.  This  may  be  due  to  the  fact  that,  as  indicated 
above,  previous  topographical  surveys  have  rendered  unnecessary 
the  "  exploratory  "  survey  which  is  required  in  a  new  country, 
and  that  their  reconnoissance  really  corresponds  more  nearly  to 
our  preliminary. 

The  perfection  to  which  barometrical  methods  have  been 
brought  has  rendered  it  possible  to  determine  differences  of 
elevation  with  sufficient  accuracy  for  reconnoissance  purposes 
by  the  combined  use  of  a  mercurial  and  an  aneroid  barometer. 
The  mercurial  barometer  should  be  kept  at  "  headquarters,"  and 
readings  should  be  taken  on  it  at  such  frequent  intervals  that 
any  fluctuation  is  noted,  and  throughout  the  period  that  observa- 
tions with  the  aneroid  are  taken  in  the  field.  At  each  observa. 
tion  there  should  also  be  recorded  the  time,  the  reading  of  the 
attached  thermometer,  and  the  temperature  of  the  external 
air.  For  uniformity,  the  mercurial  readings  should  then  be 
"  reduced  to  32°  F."  Before  starting  out,  a  reading  of  the 
aneroid  should  be  taken  at  headquarters  coincident  with  a  read- 
ing of  the  mercurial.  The  difference  is  one  value  of  the  correc- 
tion to  the  aneroid.  -  As  soon  as  the  aneroid  is  brought  back 
another  comparison  of  readings  should  be  made.  Even  though 
there  has  been  considerable  rise  or  fall  of  pressure  in  the  interval, 
the  difference  in  readings  (the  correction)  should  be  substantially 
the  same  provided  the  aneroid  is  a  good,  instrument.  The  best 
aneroids  read  directly  to  yi-g-  of  an  inch  of  mercury  and  may  be 
estimated  to  y-gVo  of  an  inch — which  corresponds  to  about  0.9 
foot  difference  of  elevation.  In  the  field  there  should  be  read, 


§  8.  RAILROAD  SURVEYS.  7 

at  each  point  whose  elevation  is  desired,  the  aneroid,  the  time, 
and  the  temperature.  These  readings,  corrected  by  the  mean 
value  of  the  correction  between  the  aneroid  and  the  mercurial, 
should  then  be  combined  with  the  reading  of  the  mercurial 
(interpolated  if  necessary)  for  the  times  of  the  aneroid  observa- 
tions and  the  difference  of  elevation  obtained.  [See  the  author's 
4 'Problems  in  the  Use  and  Adjustment  of  Engineering  In- 
struments," Prob.  22.]  Important  points  should  be  observed 
more  than  once  if  possible.  Such  duplicate  observations  will  be 
found  to  give  surprisingly  concordant  results  even  when  a 
general  fluctuation  of  atmospheric  pressure  so  modifies  the 
tabulated  readings  that  an.  agreement  is  not  at  first  apparent. 
Variations  of  pressure  produced  by  high  winds,  thunder-storms, 
«tc.,  will  generally  vitiate  possible  accuracy  by  this  method. 
By  '  '  headquarters ' '  is  meant  any  place  whose  elevation  above 
any  given  datum  is  known  and  where  the  mercurial  may  be 
placed  arid  observed  while  observations  within  a  range  of  several 
miles  are  made  with  the  aneroid.  If  necessary  the  elevation  of 
a  new  headquarters  may  be  determined  by  the  above  method, 
but  there  should  be  if  possible  several  independent  observations 
whose  accordance  will  give  a  fair  idea  of  their  accuracy. 

The  above  method  should  be  neither  slighted  nor  used  for 
more  than  it  is  worth.  When  properly  used,  the  errors  are 
compensating  rather  than  cumulative.  When  used,  for  example, 
to  determine  that  a  pass  B  is  260  feet  higher  than  a  determined 
bridge  crossing  at  A.  which  is  six  miles  distant,  and  that  another 
pass  C  is  310  feet  higher  than  A  and  is  ten  miles  distant,  the 
figures,  even  with  all  necessary  allowances  for  inaccuracy,  will 
give  an  engineer  a  good  idea  as  to  the  choice  of  route  especially 
as  affected  by  ruh'ng  grade.  There  is  no  comparison  between 
the  time  and  labor  involved  in  obtaining  the  above  information 
by  barometric  and  by  spirit-leveling  methods,  and  for  recon- 
naissance purposes  the  added  accuracy  of  the  spirit-leveling 
method  is  hardly  worth  its  cost. 

8.  Horizontal  measurements,  bearings,  etc.  When  there  is 
no  map  which  may  be  depended  on,  or  when  only  a  skeleton 


8  RAILROAD   CONSTRUCTION.  §  9. 

map  is  obtainable,  a  rapid  survey,  sufficiently  accurate  for  the 
purpose,  may  be  made  by  using  a  pocket  compass  for  bearings 
and  a  telemeter,  odometer,  or  pedometer  for  distances.  The. 
telemeter  [stadia]  is  more  accurate,  but  it  requires  a  definite  clear 
sight  from  station  to  station,  which  may  be  difficult  through  a, 
wooded  country.  The  odometer,  which  records  the  revolutions, 
of  a  wheel  of  known  circumference,  may  be  used  even  in  rough 
and  wooded  country,  and  the  results  may  be  depended  on  to  a 
small  percentage.  The  pedometer  (or  pace -measurer)  depends 
for  its  accuracy  on  the  actual  movement  of  the  mechanism  for 
each  pace  and  on  the  uniformity  of  the  pacing.  Its  results  are 
necessarily  rough  and  approximate,  but  it  may  be  used  to  fill 
in  some  intermediate  points  in  a  large  skeleton  map.  A  hand- 
ievel  is  also  useful  in  determining  the  relative  elevation  of  various 
topographical  features  which  may  have  some  bearing  on  the 
proper  location  of  the  road. 

9.  Importance  of  a  good  reconnoissance.     The  foregoing  in- 
struments and   methods   should   be  considered   only  as   aids  in 
exercising  an  educated  common  sense,  without  which  a  proper 
location  cannot  be  made.    The  reconnoissance  survey  should  com- 
mand the    best    talent  and    the    greatest   experience   available. 
If  the  general  route  is  properly  chosen,   a   comparatively  low 
order  of  engineering  skill  can  fill  in  a  location  which  will  prove 
a  paying  railroad  property ;  but  if  the  general  route  is  so  chosen 
that  the  ruling  grades  are  high  and  the  business  obtained  is  small 
and  subject  to  competition,  no  amount  of  perfection  in  detailed 
alignment  or  roadbed  construction  can  make  the  road  a  profitable 
investment. 

PRELIMINARY    SURVEYS. 

10,  Character  of  survey.      A  preliminary  railroad  survey  is 
properly  a  topographical  survey  of  a  belt  of  country  which  has 
been  selected  during  the  reconnoissance  and  within  which  it  is 
estimated  that  the  located  line  will  lie.     The  width  of  this  belt 
will  depend  on  the  character  of  the  country.     When  a  railroad 


11 


RAILROAD  SURVEYS. 


is  to  follow  a  river  having  very  steep  banks  the  choice  of  loca- 
tion is  sometimes  limited  at  places  to  a  very  few  feet  of  width 
and  the  belt  to  be  surveyed  may  be  correspondingly  narrowed. 
In  very  flat  country  the  desired  width  may  be  only  limited  by  the 
ability  to  survey  points  with  sufficient  accuracy  at  a  considerable 
distance  from  what  may  be  called  the  ' '  backbone  line  ' '  of  the 
survey. 

11.  Cross-section  method..  This  is  the  only  feasible  method 
in  a  wooded  country,  and  is  employed  by  many  for  all  kinds 
of  country.  The  backbone  line  is  surveyed  either  by  observ- 
ing magnetic  bearings  with  a  compass  or  by  carrying  forward 


FIG.  4. 


absolute  azimuths  with  a   transit.     The    compass   method   has 
the  disadvantages   of  limited    accuracy    and    the  possibility  of 


10  RAILROAD  CONSTRUCTION.  §  12. 

considerable  local  error  owing  to  local  attraction.  On  the  other 
hand  there  are  the  advantages  of  greater  simplicity,  no  necessity 
for  a  back  rodman,  and  the  fact  that  the  errors  are  purely 
local  and  not  cumulative,  and  may  be  so  limited,  with  care,  that 
they  will  cause  no  vital  error  in  the  subsequent  location  survey. 
The  transit  method  is  essentially  more  accurate,  but  is  liable 
to  be  more  laborious  and  troublesome.  If  a  large  tree  is 
encountered,  either  it  must  be  cut  down  or  a  troublesome  opera- 
tion of  offsetting  must  be  used.  If  the  compass  is  employed 
under  these  circumstances,  it  need  only  be  set  up  on  the  far  side 
of  the  tree  and  the  former  bearing  produced.  An  error  in 
reading  a  transit  azimuth  will  be  carried  on  throughout  the 
survey.  An  error  of  only  five  minutes  of  arc  will  cause  an  off- 
set of  nearly  eight  feet  in  a  mile.  Large  azimuth  errors  may, 
however,  be  avoided  by  immediately  checking  each  new  azimuth 
with  a  needle  reading.  It  is  advisable  to  obtain  true  azimuth 
at  the  beginning  of  the  survey  by  an  observation  on  the  sun  or 
Polaris,  and  to  check  the  azimuths  every  few  miles  by  azimuth 
observations.  Distances  along  the  backbone  line  should  be 
measured  with  a  chain  or  steel  tape  and  stakes  set  every  100 
feet.  When  a  course  ends  at  a  substation,  as  is  usually  the  case, 
the  remaining  portion  of  the  100  feet  should  be  measured  along 
the  next  course.  The  level  party  should  immediately  obtain  the 
elevations  (to  the  nearest  tenth  of  a  foot)  of  all  stations,  and  also 
of  the  lowest  points  of  all  streams  crossed  and  even  of  dry  gullies 
which  would  require  culverts. 

12.  Cross-sectioning.  It  is  usually  desirable  to  obtain  con- 
tours at  five-foot  intervals.  This  may  readily  be  done  by  the 
use  of  a  Locke  level  (wliich  should  be  held  on  top  of  a  simple 
five-foot  stick),  a  tape,  and  a  rod  ten  feet  in  length  graduated 
to  feet  and  tenths.  The  method  of  use  may  perhaps  be  best 
explained  by  an  example.  Let  Fig.  5  represent  a  section  per- 
pendicular to  the  survey  line — such  a  section  as  would  be  made 
by  the  dotted  lines  in  Fig.  4.  O  represents  the  station  point. 
Its  elevation  as  determined  by  the  level  is,  say,  158.3  above 
datum.  "When  the  Locke  level  on  its  five-foot  rod  is  placed  at 


§12. 


RAILROAD  SURVEYS. 


11 


(7,  the  level  has  an  elevation  of  163.3.    Therefore  when  a  point 
is  found  (as  at  a)  where  the  level  will  read  3.3  on  the  rod,  that 


FIG.  5. 


point  has  an  elevation  of  160.0  and  its  distance  from  the  center 
gives  the  position  of  the  160-foot  contour.  Leaving  the  long 
rod  at  that  point  (a),  carry  the  level  to  some  point  (5)  such  that 
the  level  will  sight  at  the  top  of  the  rod.  b  is  then  on  the  165- 


FIG.  6. 

foot  contour,  and  the  horizontal  distance  ab  added  to  the  hori- 
zontal distance  ac  gives  the  position  of  that  contour  from  the 
center.  The  contours  on  the  lower  side  are  found  similarly. 
The  first  rod  reading  will  be  8.3,  giving  the  155-foot  contour. 
Plot  the  results  in  a  note-book  which  is  ruled  in  quarter-inch 
squares,  using  a  scale  of  100  feet  per  inch  in  both  directions. 


12  RAILROAD  CONSTRUCTION.  §  13. 

Plot  the  work  UP  the  page ;  then  when  looking  ahead  along  the 
line,  the  work  is  properly  oriented.  When  a  contour  crosses 
the  survey  line,  the  place  of  crossing  may  be  similarly  deter- 
mined. If  the  ground  flattens  out  so  that  five-foot  contours  are 
very  far  apart,  the  absolute  elevations  of  points  at  even  fifty- 
foot  distances  from  the  center  should  be  determined.  The 
method  is  exceedingly  rapid.  Whatever  error  or  inaccuracy 
occurs  is  confined  in  its  effect  to  the  one  station  where  it  occurs. 
The  work  being  thus  plotted  in  the  field,  unusually  irregular 
topography  may  be  plotted  with  greater  certainty  and  no  great 
error  can  occur  without  detection.  It  would  even  be  possible 
by  this  method  to  detect  a  gross  error  that  might  have  been 
made  by  the  level  party. 

13.  Stadia  method.      This  method  is  best  adapted  to  fairly 
open  country  where   a  "shot"  to   any  desired   point   may  be 
taken  without  clearing.     The  backbone  survey  line  is  the  same 
as  in  the  previous  method  except  that  each  course  is  limited  to 
the  practicable  length  of  a  stadia  sight.     The  distance  between 
stations  should  be  checked  by  foresight  and  backsight — also  the 
vertical  angle.      Azimuths  should  be   checked  by  the  needle. 
Considering  the  vital  importance  of  leveling  on  a  railroad  survey 
it  might  be  considered  desirable  to  run  a  line  of  levels  over  the 
stadia  stations  in  order  that  the  leveling  may  be  as  precise  as 
possible ;  but  when  it  is  considered  that  a  preliminary  survey  is 
a  somewhat  hasty  survey  of  a  route  that  may  be  abandoned,  and 
that  the  errors  of  leveling  by  the  stadia  method  (which  are  com- 
pensating) may  be  so  minimized  that  no  proposed  route  would 
be  abandoned  on  account  of  such  small  error,  and  that  the  effect 
of  such  an  error  may  be  easily  neutralized  by  a  slight  change  in 
the  location,  it  may  be  seen  that  excessive  care  in  the  leveling 
of  the  preliminary  survey  is  hardly  justifiable. 

Since  the  students  taking  this  work  are  assumed  to  be  familiar 
with  the  methods  of  stadia  topographical  surveys,  this  part  of 
the  subject  will  not  be  further  elaborated. 

14.  "First"  and  "second"  preliminary  surveys.     Some  engi- 
neers advocate  two  preliminary  surveys.     When  this  is  done, 


§  15.  RAILROAD  SURVEYS.  13 

the  first  is  a  very  rapid  survey,  made  perhaps  with  a  compass, 
and  is  only  a  better  grade  of  reconnoissance.  Its  aim  is  to 
rapidly  develop  the  facts  which  will  decide  for  or  against  any 
proposed  route,  so  that  if  a  route  is  found  to  be  unfavorable 
another  more  or  less  modified  route  may  be  adopted  without 
having  wasted  considerable  time  in  the  survey  of  useless  details. 
By  this  time  the  student  should  have  grasped  the  fundamental 
idea  that  both  the  reconnoissance  and  preliminary  surveys  are 
not  surveys  of  lines  but  of  areas ;  that  their  aim  is  to  survey 
only  those  topographical  features  which  would  have  a  deter- 
mining influence  on  any  railroad  line  which  might  be  constructed 
through  that  particular  territory,  and  that  the  value  of  a  locating 
engineer  is  largely  measured  by  his  ability  to  recognize  those 
determining  influences  with  the  least  amount  of  work  from  his 
surveying  corps.  Frequently  too  little  time  is  spent  on  the 
•comparative  study  of  preliminary  lines.  A  line  will  be  hastily 
decided  on  after  very  little  study ;  it  will  then  be  surveyed  with 
minute  detail  and  estimates  carefully  worked  up,  and  the  claims 
of-  any  other  suggested  route  will  then  be  handicapped,  if  not 
disregarded,  owing  to  an  unwillingness  to  discredit  and  throw 
away  a  large  amount  of  detailed  surveying.  The  cost  of  two  or 
three  extra  preliminary  surveys  (at  critical  points  and  not  over 
the  whole  line)  is  utterly  insignificant  compared  with  the  prob- 
able improvement  in  the  "  operating  value"  of  a  line  located 
after  such  a  comparative  study  of  preliminary  lines. 

LOCATION    SURVEYS. 

15.  "Paper  location,"  "When  the  preliminary  survey  has 
been  plotted  to  a  scale  of  200  feet  per  inch  and  the  contours 
drawn  in,  a  study  may  be  made  for  the  location  survey.  Disre- 
garding for  the  present  the  effect  on  location  of  transition  curves, 
the  alignment  may  be  said  to  consist  of  straight  lines  (or  ' '  tan- 
gents ")  and  circular  curves.  The  "  paper  location  "  therefore 
consists  in  plotting  on  the  preliminary  map  a  succession  of 
straight  lines  which  are  tangent  to  the  circular  curves  connect- 


14  RAILROAD  CONSTRUCTION.  §  15. 

ing  them.  The  determining  points  should  first  be  considered. 
Such  points  are  the  termini  of  the  road,  the  lowest  practicable 
point  over  a  summit,  a  river-crossing,  etc.  So  far  as  is  possi- 
ble, having  due  regard  to  other  considerations,  the  road  should 
be  a  "surface"  road,  i.e.,  the  cut  and  fill  should  be  made  as 
small  as  possible.  The  maximum  permissible  grade  must  also 
have  been  determined  and  duly  considered.  The  method  of 
location  differs  radically  according  as  the  lines  joining  the  deter- 
mining points  have  a  very  low  grade  or  have  a  grade  that  ap- 
proaches the  maximum  permissible.  With  very  low  natural 
grades  it  is  only  necessary  to  strike  a  proper  balance  between 
the  requirements  for  easy  alignment  and  the  avoidance  of  exces- 
sive earthwork.  When  the  grade  between  two  determined 
points  approaches  the  maximum,  a  study  of  the  location  may  be 
begun  by  finding  a  strictly  surface  line  which  will  connect  those 
points  with  a  line  at  the  given  grade.  For  example,  suppose 
the  required  grade  is  1.6$  and  that  the  contours  are  drawn  at 
5-foot  intervals.  It  will  require  312  feet  of  1.6$  grade  to  rise 
5  feet.  Set  a  pair  of  dividers  at  312  feet  and  step  off  this  in- 
terval on  successive  contours.  This  line  will  in  general  be  very 
irregular,  but  in  an  easy  country  it  may  lie  fairly  close  to  the 
proper  location  line,  and  even  in  difficult  country  such  a  surface 
line  will  assist  greatly  in  selecting  a  suitable  location.  When  the 
larger  part  of  the  line  will  evidently  consist  of  tangents,  the  tan- 
gents should  be  first  located  anfl  should  then  be  connected  by 
suitable  curves.  When  the  curves  predominate,  as  they  gener- 
ally will  in  mountainous  country,  and  particularly  when  the  line 
is  purposely  lengthened  in  order  to  reduce  the  grade,  the  curves 
should  be  plotted  first  and  the  tangents  may  then  be  drawn 
connecting  them.  Considering  the  ease  with  which  such  lines 
may  be  drawn  on  the  preliminary  map,  it  is  frequently  advisable, 
after  making  such  a  paper  location,  to  begin  all  over,  draw  a 
new  line  over  some  specially  difficult  section  and  compare  re- 
sults. Profiles  of  such  lines  may  be  readily  drawn  by  noting  their 
intersection  with  each  contour  crossed.  Drawing  on  each  profile 
the  required  grade  line  will  furnish  an  approximate  idea  of  the 


§  16.  RAILROAD  SURVEYS.  15 

comparative  amount  of  earthwork  required.  After  deciding  on 
the  paper  location,  the  length  of  each  tangent,  the  central  angle 
(see  §  21),  and  the  radius  of  each  curve  should  be  measured  as 
accurately  as  possible.  Since  a  slight  error  made  in  such  meas- 
urements, taken  from  a  map  with  a  scale  of  200  feet  per  inch, 
would  by  accumulation  cause  serious  discrepancies  between  the 
plotted  location  and  the  location  as  afterward  surveyed  in  the 
field,  frequent  tie  lines  and  angles  should  be  determined  between 
the  plotted  location  line  and  the  preliminary  line,  and  the  loca- 
tion should  be  altered,  as  may  prove  necessary,  by  changing  the 
length  of  a  tangent  or  changing  the  central  angle  or  radius  of  a 
curve,  so  that  the  agreement  of  the  check-points  will  be  suffi- 
ciently close.  The  errors  of  an  inaccurate  preliminary  survey 
may  thus  be  easily  neutralized  (see  §  33).  When  the  pre- 
liminary line  has  been  properly  run,  its  "  backbone  "  line  will 
lie  very  near  the  location  line  and  will  probably  cross  it  at  fre- 
quent intervals,  thus  rendering  it  easy  to  obtain  short  and  nu- 
merous tie  lines. 

16.  Surveying  methods.  A  transit  should  be  used  for  align- 
ment, and  only  precise  work  is  allowable.  The  transit  stations 
should  be  centered  with  tacks  and  should  be  tied  to  witness- 
stakes,  which  should  be  located  outside  of  the  range  of  the  earth- 
work, so  that  they  will  neither  be  dug  up  nor  covered  up.  All 
original  property  lines  lying  within  the  limits  of  the  right  of  way 
should  be  surveyed  with  reference  to  the  location  line,  so  that 
the  right-of-way  agent  may  have  a  proper  basis  for  settlement. 
" W  hen  the  property  lines  do  not  extend  far  outside  of  the  re- 
quired right  of  way  they  are  frequently  surveyed  completely. 

The  leveler  usually  reads  the  target  to  the  nearest  thousandth 
of  a  foot  on  turning-points  and  bench-marks,  but  reads  to  the 
nearest  tenth  of  a  foot  for  the  elevation  of  the  ground  at 
stations.  Considering  that  y^f^  of  a  foot  has  an  angular  value 
of  only  7  seconds  at  a  distance  of  300  feet,  and  that  one  division 
of  a  level-bubble  is  usually  about  30  seconds,  it  may  be  seen  that 
it  is  a  useless  refinement  to  read  to  thousandths  unless  corre- 
sponding care  is  taken  in  the  use  of  the  level.  The  leveler 


16  RAILROAD  CONSTRUCTION.  §  17. 

should  also  locate  his  bench-marks  outside  of  the  range  of 
earthwork.  A  knob  of  rock  protruding  from  the  ground  affords 
an  excellent  mark.  A  large  nail,  driven  in  the  roots  of  a  tree, 
which  is  not  to  be  disturbed,  is  also  a  good  mark.  These  marks 
should  be  clearly  described  in  the  note-book.  The  leveler  should 
obtain  the  elevation  of  the  ground  at  all  station-points ;  also  at 
all  sudden  breaks  in  the  profile  line,  determining  also  the  distance 
of  these  breaks  from  the  previous  even  station.  This  will  in- 
clude the  position  and  elevation  of  all  streams,  and  even  dry 
gullies,  which  are  crossed. 

Measurements  should  preferably  be  made  with  a  steel  tape, 
care  being  taken  on  steep  ground  to  insure  horizontal  measure- 
ments. Stakes  are  set  each  100  feet,  and  also  at  the  beginning 
and  end  of  all  curves.  Transit-points  (sometimes  called  ' '  plugs  ' ' 
or  "hubs")  should  be  driven  flush  with  the  ground,  and  a 
*c  witness- stake,"  having  the  "number"  of  the  station,  should 
be  set  three  feet  to  the  right.  For  example,  the  witness-stake 
might  have  on  one  side  "  137  -f-  69.92,"  and  on  the  other  side 
"  P  C  4°  K,"  which  would  signify  that  the  transit  hub  is  69.92 
feet  beyond  station  137,  or  13769.92  feet  from  the  beginning  of 
the  line,  and  also  that  it  is  the  "point  of  curve"  of  a  "4°- 
curve ' '  which  turns  to  the  right. 

Alignment.     The  alignment  is  evidently  a  part  of  the  loca- . 
tion  survey,  but,  on  account  of  the  magnitude  and  importance 
of  the  subject,  it  will  be  treated  in  a  separate  chapter. 

17.  Form  of  Notes.  Although  the  Form  of  Notes  cannot  be 
thoroughly  understood  until  after  curves  are  studied,  it  is  nere 
introduced  as  being  the  most  convenient  place.  The  right-hand 
page  should  have  a  sketch  showing  all  roads,  streams,  and 
property  lines  crossed  with  the  bearings  of  those  lines.  This 
should  be  drawn  to  a  scale  of  100  feet  per  inch — the  quarter- 
inch  squares  which  are  usually  ruled  in  note-books  giving  con- 
venient 25-foot  spaces.  This  sketch  will  always  be  more  or  less 
distorted  on  curves,  since  the  center  line  is  always  shown  as 
straight  regardless  of  curves.  The  station  points  ("Sta."  in 
first  column,  left-hand  page)  should  be  placed  opposite  to  their 


§17. 


RAILROAD  SURVEYS. 


17 


sketched  positions,  which  means  that  even  stations  will  be 
recorded  on  every  fourth  line.  This  allows  three  intermediate 
lines  for  substations,  which  is  ordinarily  more  than  sufficient. 
The  notes  should  read  UP  the  page,  so  that  the  sketch  will  be 
properly  oriented  when  looking  ahead  along  the  line.  The 
other  columns  on  the  left-hand  page  will  be  self-explanatory 
when  the  subject  of  curves  is  understood.  If  the  ' '  calculated 
bearings ' '  are  based  on  azimuthal  observations,  their  agreement 
(or  constant  difference)  with  the  needle  readings  will  form  a 
valuable  check  on  the  curve  calculations  and  the  instrumental 
work. 


FORM  OF  NOTES. 


[Left-hand  page.] 


[Right-hand  page.] 


Sta. 


54 


53 
Q+72.2 

52 


O      50 


48 

0+32 
47 


Align- 
ment 


gg 

* 


P.O. 


Vernier 


9°  11' 
7  57 

6  15 
4  33 
2  51 


09 


0° 


18°  22' 


Calc. 
Bearing. 


N  54°  48'  E 


N  36"  26'  E 


Needle. 


N  62°  15'  1 


N44°    O't 


52+18 


WM.  BROWN 


JOHN    JONES 


46+31 


\ 


CHAPTEE  II. 

ALIGNMENT. 

IN  this  chapter  the  alignment  of  the  center  line  only  of  a 
pair  of  rails  is  considered.  When  a  railroad  is  crossing  a  sum- 
mit in  the  grade  line,  although  the  horizontal  projection  of  the 
alignment  may  be  straight,  the  vertical  projection  will  consist  of 
two  sloping  lines  joined  by  a  curve.  When  a  curve  is  on  a 
grade,  the  center  line  is  really  a  spiral,  a  curve  of  double  curva- 
ture, although  its  horizontal  projection  is  a  circle.  The  center 
line  therefore  consists  of  straight  lines  and  curves  of  single 
and  double  curvature.  The  simplest  method  of  treating  them 
is  to  consider  their  horizontal  and  vertical  projections  separately. 
In  treating  simple,  compound,  and  transition  curves,  only  the 
horizontal  projections  of  those  curves  will  be  considered. 


SIMPLE    CUKVES. 

18.  Designation  of  curves.  A  curve  may  be  designated 
either  by  its  radius  or  by  the  angle 
subtended  by  a  chord  of  unit  length. 
Such  an  angle  is  known  as  the  ' c  degree 
of  curve ' '  and  is  indicated  by  D. 
Since  the  curves  that  are  practically 
used  have  very  long  radii,  it  is  gener- 
ally impracticable  to  make  any  use  of 
the  actual  center,  and  the  curve  is 
located  without  reference  to  it.  If 
AB  in  Fig.  7  represents  a  unit  chord 
(C)  of  a  curve  of  radius  7?,  then  by  the  above  defini- 

18 


§  19.  ALIGNMENT.  19 

tion  the  angle  AOB  equals  D.     Then  AO  sin  \D  =  \AB  = 


sin 


or,  by  inversion, 


(1) 


The  unit  chord  is  variously  taken  throughout  the  world  as 
100  feet,  66  feet,  and  20  meters.  In  the  United  States  100 
feet  is  invariably  used  as  the  unit  chord  length,  and  throughout 
this  work  it  will  be  so  considered.  Table  I  has  been  computed 
on  this  basis.  It  gives  the  radius,  with  its  logarithm,  of  all 
curves  from  a  0°  01'  curve  up  to  a  10°  curve,  varying  by  single 
minutes.  The  sharper  curves,  which  are  seldom  used,  are  given 
with  larger  intervals. 

An  approximate  value  of  R  may  be  readily  found  from  the 
following  simple  rule,  which  should  be  memorized : 

5730 
^~    D  ' 

Although  such  values  are  not  mathematically  correct,  since  R 
does  not  strictly  vary  inversely  as  D,  yet  the  resulting  value  is 
within  a  tenth  of  one  per  cent  for  all 
commonly  used  values  of  7?,  and  is  suf- 
ficiently close  for  many  purposes,  as  will 
be  shown  later. 

19.  Length  of  a  sub-chord.  Since 
it  is  impracticable  to  measure  along  a 
curved  arc,  curves  are  always  measured 
by  laying  off  100-foot  chord  lengths. 
This  means  that  the  actual  arc  is  always 
a  little  longer  than  the  chord.  It  also 
means  that  a  subchord  (a  chord  shorter  than  the  unit  length) 
will  be  a  little  longer  than  the  ratio  of  the  angles  subtended 
would  call  for.  The  truth  of  this  may  be  seen  without  calcu- 


A     ' 


FIG.  8. 


20  RAILROAD   CONSTRUCTION.  §  20. 

lation  by  noting  that  two  equal  subchords,  each  subtending  the 
angle  \D,  will  evidently  be  slightly  longer  than  50  feet  eaclu 
If  c  be  the  length  of  a  subchord  subtending  the  angle  d,  then, 
as  in  Eq.  (2), 


or,  by  inversion, 

c=2.R  sin  %d.    .     .....     (3) 

The  nominal  length  of  a  subchord  =  100—.     For  example, 


a  nominal  subchord  of  40  feet  will  subtend  an  angle  of  -^Q  of 
D°  ;  its  true  length  will  be  slightly  more  than  40  feet,  and  may 
be  computed  by  Eq.  3.  The  difference  between  the  nominal 
and  true  lengths  is  maximum  when  the  subchord  is  about  57 
feet  long,  but  with  the  low  degrees  of  curvature  ordinarily  used 
the  difference  may  be  neglected.  With  a  10°  curve  and  a 
nominal  chord  length  of  60  feet,  the  true  length  is  60.049  feet. 
Yery  sharp  curves  should  be  laid  off  with  50  -foot  or  even  25- 
foot  chords  (nominal  length).  In  such  cases  especially  the  true 
lengths  of  these  subcliords  should  be  computed  and  used  instead 
of  the  nominal  lengths. 

20.  Length  of  a  curve.  The  length  of  a  curve  is  always 
indicated  by  the  quotient  of  100^  -=-  D.  If  the  quotient  of 
A  ~  D  is  a  whole  number,  the  length  as  thus  indicated  is  the 
true  length  —  measured  in  100-foot  chord  lengths.  If  it  is  an 
odd  number  or  if  the  curve  begins  and  ends  with  a  subchord 
(even  though  A  -f-  D  is  a  whole  number),  theoretical  accuracy 
requires  that  the  true  subchord  lengths  shall  be  used,  although 
the  difference  may  prove  insignificant.  The  length  of  the  arc 
(or  the  mean  length  of  the  two  rails)  is  therefore  always  in 
excess  of  the  length  as  given  above.  Ordinarily  the  amount 
of  this  excess  is  of  no  practical  importance.  It  simply  adds  an 
insignificant  amount  to  the  length  of  rail  required. 

Example.  Required  the  nominal  and  true  lengths  of  a 
3°  45'  curve  having  a  central  angle  of  17°  25'.  First  reduce 


§22. 


ALIGNMENT. 


21 


the  degrees  and  minutes  to  decimals  of  a  degree.  (100  X  17°  25') 
^  3°  45'  =  1741.667  -5-  3.75  =  464.444.  The  curve  has  four 
100-foot  chords  and  a  nominal  chord  of  64.444.  The  true 
chord  should  be  64.451.  The  actual  arc  is 


17°.4167  X  ^      o  X 
loO 


=  464.527. 


The  excess  is  therefore  464.527  —  464.451  =  0.076  foot. 

21.  Elements  of  a  curve.  Considering  the  line  as  running 
from  A  toward  B,  the  beginning  of  the  curve,  at  A,  is  called 
\\IQ  point  of  curve  (PC).  The  other  end  of  the  curve,  at  .Z?,  is 
called  the  point  of  tangency  (PT). 
The  intersection  of  the  tangents  is 
called  the  vertex  (V).  The  angle 
made  by  the  tangents  at  F,  which 
equals  the  angle  made  by  the  radii  to 
the  extremities  of  the  curve,  is  called 
the  central  angle  (A).  A  Fand  B  F, 
the  two  equal  tangents  from  the  vertex 
to  the  PC  and  PT,  are  called  the 
tangent  distances  (T).  The  chord 
AB  is  called  the  long  chord  (LC). 
The  intercept  HG  from  the  middle 
of  the  long  chord  to  the  middle  of  the  arc  is  called  the  middle 
ordinate  (M).  That  part  of  the  secant  #Ffrom  the  middle  of 
the  arc  to  the  vertex  is  called  the  external  distance  (E).  From 
the  figure  it  is  very  easy  to  derive  the  following  frequently  used 
relations : 


FIG.  9. 


(4) 

(5) 


M=  R 

E  =  R  exsec  | A 


22.  Relation  between  T,  E,  and  A.     Join  A  and  G  in  Fig.  9. 

The  angle  VAG  =  \A,  since  it  is  measured  by  one  half  of  the 


22  RAILROAD  CONSTRUCTION.  §  23. 

arc  AG  between  the  secant  and  tangent.     AGO  =  90°  —  JJ. 

J.F:  F#::sin  J.#F:sin  VAG\ 
sin  J.#  F=  sin  AGO  —  cos iJ ; 

T:  ^  :  :  cos  J  J  :  sin  i  J  ; 
T^^cotiJ (8) 

The  same  relation  may  be  obtained  by  dividing  Eq.  4  by  Eq.  7, 
since  tan  a  -f-  exsec  a  —  cot  \a. 

23.  Elements  of  a  1°  curve.     From  Eqs.  1  to  8  it  is  seen  that 
the  elements  of  a  curve  vary  directly  as  R.     It  is  also  seen  to 
be  very  nearly  true  that  R  varies  inversely  as  D.     If  the  ele- 
ments of  a  1°  curve  for  various  central  angles  are  calculated  and 
tabulated,   the  elements  of  a  curve  of  D0  curvature   may  be 
approximately  found  by  dividing  by  D  the  corresponding  elements 
of  a  1°  curve  having  the  same  central  angle.     For  small  central 
angles  and  low  degrees  of  curvature  the  errors  involved  by  the 
approximation  are  insignificant,  and  even  for  larger  angles  the 
errors  are  so  small  that /or  many  purposes  they  may  be  disre- 
garded. 

In  Table  II  is  given  the  value  of  the  tangent  distances, 
external  distances,  and  long  chords  for  a  1°  curve  for  various 
central  angles.  The  student  should  familiarize  himself  with  the 
degree  of  approximation  involved  by  solving  a  large  number  of 
cases  under  various  conditions  by  the  exact  and  approximate 
methods,  in  order  that  he  may  know  when  the  approximate 
method  is  sufficiently  exact  for  the  intended  purpose.  The 
approximate  method  also  gives  a  ready  check  on  the  exact 
method. 

24.  Exercises,     (a)  What  is  the  tangent  distance  of  a  4°  20' 
curve  having  a  central  angle  of  18°  24'  ? 

(b)  Given  a  3°   30'  curve  and  a  central  angle  of  16°  20', 
how  far  will  the  curve  pass  from  the  vertex  ?     [Use  Eq.  7.] 

(c)  An  18°  curve  is  to  be  laid  off  using  25-foot  (nominal) 
chord  lengths.     What  is  the  true  length  of  the  subchords  ? 


§25. 


ALIGNMENT. 


23 


(d)  Given  two  tangents  making  a  central  angle  of  15°  24'. 
It  is  desired  to  connect  these  tangents  by  a  curve  which  shall 
pass  16.2  feet  from  their  intersection.  How  far  down  the 
tangent  will  the  curve  begin  and  what  will  be  its  radius  ?  (Use 
Eq.  8  and-then  use  Eq.  4  inverted.) 

25.  Curve  location  by  deflections,  The  angle  between  a 
secant  and  a  tangent  (or  between  two  secants  intersecting  on  an 
arc)  is  measured  by  one  half  of  the  intercepted  arc.  Beginning 
at  the  PC  (A  in  Fig.  10),  if  the  first  chord  is  to  be  a  full  chord 
we  may  deflect  an  angle  VAa  (=  ?D), 
and  the  point  a,  which  is  100  feet  from 
J.,  is  a  point  on  the  curve.  For  the 
next  station,  J,  deflect  an  additional 
angle  bAa  (—  %D)  and,  with  one  end 
of  the  tape  at  a,  swing  the  other  end 
until  the  100-foot  point  is  on  the  line 
Ab.  The  point  b  is  then  on  the  curve. 
If  the  final  chord  cB  is  a  subchord,  its 
additional  deflection  (^a)  is  something 
less  than  \D.  The  last  deflection 
(BA  V)  is  of  course  \A.  It  is  particularly  important,  when  a 
curve  begins  or  ends  with  a  subchord  and  the  deflections  are 
odd  quantities,  that  the  last  additional  deflection  should  be  care- 
fully computed  and  added  to  the  previous  deflection,  to  check 
the  mathematical  work  by  the  agreement  of  this  last  computed 
deflection  with  %A. 

Example.  Given  a  3°  24'  curve  having  a  central  angle  of 
18°  22'  and  beginning  at  sta.  47  -|-  32,  to  compute  the  deflections. 
The  nominal  length  of  curve  is  18°  22'  ~  3°  24'  =  IS. 367 -^ 
3.40  —  5.402  stations  or  540.2  feet.  The  curve  therefore  ends 
at  sta.  52  +  72.2.  The  deflection  for  sta.  48  is  T6F8o-  X  K3°  24') 
=  0.68  X  1°.7  =  1M56  =  1°  09'  nearly.  For  each  additional 
100  feet  it  is  1°  42'  additional.  The  final  additional  deflection 
for  the  final  subchord  of  72.2  feet  is 


FIG.  10. 


72.2 


X  1(3°  24')  =  1°.2274  =  1°  14r  nearly. 


24  RAILROAD  CONSTRUCTION.  §26. 

The  deflections  are 

P.  C....Sta.  47  +  32 0° 

48 0°         +  1°  09'  =  1°  09' 

49 1°  09'+  1°  42'  =  2°  51' 

50 2°  51'  +  1°  42'  =4°  33' 

51 4°  33' +  1°  42' =  6°  15' 

52 6°  15'  +  1°  42'  =  7°  57' 

p.  T 52  +  72.2 ....  7°  57^  +  1°  14'  =  9°  IV 

As  a  check  9°  11'  =  i(18°  22')  =  \A.  (See  the  Form  of  Notes 
in  §  17.) 

26.  Instrumental  work.  It  is  generally  impracticable  to 
locate  more  than  500  to  600  feet  of  a  curve  from  one  station. 
Obstructions  will  sometimes  require  that  the  transit  be  moved  up 
every  200  or  300  feet.  There  are  two  methods  of  setting  off 
the  angles  when  the  transit  has  been  moved  up  from  the  PC. 

(a)  The  transit  may  be  sighted  at  the  previous  transit  station 
with  a  reading  on  the  plates  equal  to  the  deflection  angle  from 
that  station  to  the  station  occupied,  but  with  the  angle  set  oft"  on 
the  oilier  side  of  0°,  so  that  when  the  telescope  is  turned  to  0°  it 
will  sight  along  the  tangent  at  the  station  occupied.  Plunging 
the  telescope,  the  forward  stations  may  be  set  off  by  deflecting 
the  proper  deflections  from  the  tangent  at  the  station  occupied. 
This  is  a  very  common  method  and,  when  the  degree  of  curva- 
ture is  an  even  number  of  degrees  and  when  the  transit  is  only 
set  at  even  stations,  there  is  but  little  objection  to  it.  But  the 
degree  of  curvature  is  sometimes  an  odd  quantity,  and  the  exi- 
gencies of  difficult  location  frequently  require  that  substations 
be  occupied  as  transit  stations.  Method  (a)  will  then  require 
the  recalculation  of  all  deflections  for  each  new  station  occupied. 
The  mathematical  work  is  largely  increased  and  the  probability 
of  error  is  very  greatly  increased  and  not  so  easily  detected. 
Method  (b)  is  just  as  simple  as  method  (a)  even  for  the  most 
simple  cases,  and  for  the  more  difficult  cases  just  referred  to  the 
superiority  is  very  great. 


§26. 


ALIGNMENT. 


(b)  Calculate  the  deflection  for  each  station  and  substation 
throughout  the  curve  as  though  the  whole  curve  were  to  be  lo- 
cated from  the  PC.  The  computations  may  thus  be  completed 
and  checked  (as  above)  before  beginning  the  instrumental  work. 
If  it  unexpectedly  becomes  necessary  to  introduce  a  substation 
at  any  point,  its  deflection  from  the  P(7may  be  readily  inter- 
polated. The  stations  actually  set  from  the  PC  are  located  as 
usual.  RULE.  When  the  transit  is  set  on  any  forward  station, 
backsight  to  ANY  previous  station  with  the  plates  set  at  the  deflec- 
tion angle  for  the  station  sighted  at.  Plunge  the  telescope  and 
sight  at  any  forward  station  with  the  deflection  angle  originally 
computed  for  that  station.  When  the  plates  read  the  deflection 
angle  for  the  station  occupied,  the  telescope  is  sighting  along  the 
tangent  at  that  station — which  is  the  method  of  getting  the  for- 
ward tangent  when  occupying  the  PT.  Even  though  the  sta- 
tion occupied  is  an  unexpected  substation, 
when  the  instrument  is  properly  oriented  at 
that  station,  the  angle  reading  for  any  station,  \ 
forward  or  back,  is  that  originally  computed 
for  it  from  the  PC.  In  difficult  work,  where 
there  are  obstructions,  a  valuable  check  on 
the  accuracy  may  be  found  by  sighting  back- 
ward at  any  visible  station  and  noting  whether 
its  deflection  agrees  with  that  originally  com- 
puted. As  a  numerical  illustration,  assume 
a  4°  curve,  with  28°  curvature,  with  stations 
0,  2,  4,  and  7  occupied.  After  setting 
stations  1  and  2,  set  up  the  transit  at  sta. 
2  and  backsight  to  sta.  0  with  the  deflection 
for  sta.  0,  which  is  0°.  The  reading  on  sta. 
1  is  2° ;  when  the  reading  is  4°  the  telescope 
is  tangent  to  the  curve,  and  when  sighting 
at  3  and  4  the  deflections  will  be  6°  and  8°. 
Occupy  4 ;  sight  to  2  with  a  reading  of  4( 
is  8°  the  telescope  is  tangent  to  the  curve  and,  by  plunging  the 
telescope,  5,  6,  and  7  may  be  located  with  the  originally  com- 


FIG.  11. 


When  the  reading 


RAILROAD  CONSTRUCTION. 


§27 


puted  deflections  of   10°,  12°,  and  14°.     When  occupying  7  a 
backsight  may  be  taken  to  any  visible  station  with  the  plates  read 
irig  the  deflection  for  that  station ;    then  when  the  plates  read 
14:°  the  telescope  will  point  along  the  forward  tangent. 

The  location  of  curves  by  deflection  angles  is  the  normal 
method.  A  few  other  methods,  to  be  described,  should  be  con- 
sidered as  exceptional. 

27.  Curve  location  by  two  transits.  A  curve  might  be  located 
more  or  less  on  a  swamp  where  accurate  chaining  would  be  ex- 
ceedingly difficult  if  not  impossible.  The  long  chord  AB  may 
be  determined  by  triangulation  or  otherwise,  and  the  elements  of 


FIG.  12. 


FIG.  13. 


the  curve  computed,  including  (possibly)  subchords  at  each  end. 
The  deflection  from  A  and  B  to  each  point  may  be  computed. 
A  rodman  may  then  be  sent  (by  whatever  means)  to  locate  long 
stakes  at  points  determined  by  the  simultaneous  sightings  of  the 
two  transits. 

28.  Curve  location  by  tangential  offsets.      When  a  curve  is 
very  flat  and  no  transit  is  at  hand  the  following  method  may  be 


(9) 


§  29.  ALIGNMENT.  27 

used :  Produce  the  back  tangent  as  far  forward  as  necessary. 
Compute  the  ordinates  Oa' ,  Ob' ,  Oc ',  etc.,  and  the  abscissae  a! a, 
Vb,  c'c,  etc.  If  Oa  is  a  full  station  (100  feet),  then 

Oa'  =  Oa'  =  100  cos  \D,    also  =  R  sin  D ; 

Ob'  =  Oa'  +  a'V  =  100  cos  1$D+  100  cos  f  Z>, 

also  =  R  sin  2Z> ; 

also  —  R  sin  3D ; 
etc. 

a' a  =  100  sin  £Z>,  also  =-J2v-ersZ>; 

&'£  —  a'a  +  Vb  =100  sin  ^7>  +  100  sin  |  />, 

also  =  j?£vers2Z>; 

eV  =  VI  +c"c  =  100(siniZ>+sinfZ>+sinfZ>), 

also  — 
etc. 


(10) 


The  functions  JZ>,  fZ^,  etc.,  may  be  more  conveniently  used 
without  logarithms,  by  adding  the  several  natural  trigonometrical 
functions  and  pointing  off  two  decimal  places.  It  may  also  be 
noted  that  ob'  (for  example)  is  one  half  of  the  long  chord 
for  four  stations ;  also  that  b'b  is  the  middle  ordinate  for  four 
stations.  If  the  engineer  is  provided  with  tables  giving  the  long 
chords  and  middle  ordinates  for  various  degrees  of  curvature, 
these  quantities  may  be  taken  (perhaps  by  interpolation)  from 
such  tables. 

If  the  curve  begins  or  ends  at  a  substation,  the  angles  and 
terms  will  be  correspondingly  altered.  The  modifications  may 
be  readily  deduced  on  the  same  principles  as  above,  and  should 
be  worked  out  as  an  exercise  by  the  student. 

29.  Curve  location  by  middle  ordinates.  Take  first  the  simpler 
case  when  the  curve  begins  at  an  even  station.  If  we  consider 
(in  Fig.  14)  the  curve  produced  back  to  2,  the  chord  sa  =• 
2  X  100  cos  iZ>,  A' a  —  100  cos  £Z>,  and  A' A  =  am  =  zn  = 
100  sin  \D.  Set  off  A  A'  perpendicular  to  the  tangent  and 
A' a  parallel  to  the  tangent.  A  A!  =  aa'  =  W  =  <?<?',  etc.  = 
100  sin  \D.  Set  off  aa'  perpendicular  to  a' A.  Produce  Aa,' 


28 


RAILROAD  CONSTRUCTION. 


§30. 


until  a'l>  — •  A1  a,  thus  determining  5.     Succeeding  points  of  the 
curve  may  thus  be  determined  indefinitely. 

Suppose  the  curve  begins  with  a  subchord.      As   before 
ra  =  Am'  =  c'  cos    d' ,   and  rA  =  am'  =  c'  sin  \d' .     Also  sz 
and   sA  =  zn'  =  G"  sin  \d  "     in    which 


z. 


FIG.  14. 


FIG.  15. 


(<#'  -f-  <#")  =  Z>.  The  points  0  and  a  being  determined  on  the 
ground,  aa'  may  be  computed  and  set  off  as  before  and  the  curve 
continued  in  full  stations.  A  subchord  at  the  end  of  the  curve 
may  be  located  by  a  similar  process. 

30,  Curve  location  by  offsets  from  the  long  chord.  (Fig.  16.) 
Consider  at  once  the  general  case  in  which  the  curve  commences 
with  a  subchord  (curvature,  d'\  contains  with  one  or  more  full 
chords  (curvature  of  each,  /?),  and  ends  with  a  subchord  with 
curvature  d" '.  The  numerical  work  consists  in  computing  first 
AB,  then  the  various  abscissae  and  ordinates.  AB—^E  sin^J. 


Aa'=Aaf  =  c' cos$(4  -  d'); 

AV  =  Aa'  +  a'b'  =  c'  cos  \(A  -  d'}  +  r  00  cos  \(A  -  2d'  -  D] ; 

Ac'  =  Aa'  -f  a'b'  -f  Vef  =  c'  cos  \(A  -  d')  + 100  cos  \(A  -  2d'  -  D) 

+  100  cos  \(A-  W-  D); 
also 

-  c"  cosl>(J  -  d"). 


(11) 


§32. 


ALIGNMENT. 


29 


of  a—  a'  a          =  c'  sin  \(A  —  d')\ 

b'b  =  a'a  -f  mb=  c'  sin  \(A  -  d')+  100  sin  \(A  -2d'  -  I)}; 

C'C  =  vb  _  nb  =  c'  sin  \(A  -  d')+  100  sin  -|(  J  -  2d'  -  D) 


(12) 


also 


=c"  sin  £(  A  -  d"). 


The  above  formulae  are  considerably  simplified  when  the  curve 
begins  and  ends  at  even  stations.     When  the  curve  is  very  long 
a  regular  law  becomes    very    apparent   in   the 
formation  of  all  terms  between  the  first  and  last. 
There  are  too  few  terms  in  the  above  equations 
to  show  the  law. 

31.  Use  and  value  of  the  above  methods.     The 
chief  value  of  the  above    methods    lies   in   the 
possibility  of  doing  the  work  without  a  transit. 
The  same  principles  are  sometimes    employed, 
even  when  a  transit  is  used,  when  obstacles  pre- 
vent the  use  of  the  normal  method  (see  §  32,  c). 
If  the  terminal  tangents  have  already  been  ac- 
curately determined,  these  methods  are  useful  to 
locate  points  of  the  curve  when  rigid  accuracy 
is  not  essential.     Track  foremen  frequently  use 
such  methods    to  lay  out  unimportant    sidings, 

especially  when  the  engineer  and  his  transit  are  not  at  hand. 
Location  by  tangential  offsets  (or  by  offsets  from  the  long  chord) 
is  to  be  preferred  when  the  curve  is  flat  (i.e.,  has  a  small  central 
angle  ^/)  and  there  is  no  obstruction  along  the  tangent,  or  long 
chord.  Location  by  middle  ordinates  may  be  employed  regard- 
less of  the  length  of  the  curve,  and  in  cases  when  both  the 
tangents,  and  the  long  chord  are  obstructed.  The  above 
methods  are  but  samples  of  a  large  number  of  similar  methods 
which  have  been  devised.  The  choice  of  the  particular 
method  to  be  adopted  must  be  determined  by  the  local  con- 
ditions. 

32.  Obstacles  to  location.     In  this  section  will  be  given  only 
a  few  of  the  principles  involved  in  this  class  of  problems,  with 
illustrations.     The  engineer  must  decide  in  each  case,  which  ia 


FIG. 


30 


RAILROAD  CONSTRUCTION. 


32. 


the  best  method  to  use,  and  it  is  frequently  advisable  to  devise  a 
special  solution  for  some  particular  case. 

a,  When  the  vertex  is  inaccessible.  As  shown  in  §  26,  it  is 
not  absolutely  essential  that  the  vertex  of  a  curve  should  be 
located  on  the  ground.  But  it  is  very  evident  that  the  angle 
between  the  terminal  tangents  is  determined  with  far  less  prob- 
able error  if  it  is  measured  by  a  single  measurement  at  the  ver- 
tex rather  than  as  the  result  of  numerous  angle  measurements 
along  the  curve,  involving  several  positions  of  the  transit 
and  comparatively  short  sights.  Sometimes  the  location  of  the 
tangents  is  already  determined  on  the  ground  (as  by  bn  and  am, 
Fig,  17),  and  it  is  required  to  join  the  tangents  by  a  curve  of 
given  radius.  Method.  Measure  ab  and  the  angles  Vba  and 
ba  V.  A  is  the  sum  of  these  angles.  The  distances  b  V  and  a  V 
are  computable  from  the  above  data.  Given  A  and  /£,  the  tan- 


FIG.  18. 

gent  distances  are  computable,  and  then  Bb  and  aA  are  found 
by  subtracting  b  V  and  a  V  from  the  tangent  distances.  The 
curve  may  then  be  run  from  A,  and  the  work  may  be  checked 
by  noting  whether  the  curve  as  run  ends  at  B — previously  lo- 
cated from  b. 

b.  When  the  point  of  curve  (or  point  of  tangency)  is  inacces- 
sible.    At  some  distance  (As,  Fig.  18)  an  unobstructed  line  pn 


§  33.  ALIGNMENT.  31 

may  be  run  parallel  "with   A  V.     nv  =  py  =  As  =  It  vers  a. 

vers  a  =  As  -f-  It.     ns  =ps  —  It  sin  a. 

At  y,  which  is  at  a  distance  ^>s  back  from  the  computed  posi- 
tion of  A,  make  an  offset  sA  to  JP.  Eun  pn  parallel  to  the 
tangent.  A  tangent  to  the  curve  at  n  makes  an  angle  of  a  with 
np.  From  n  the  curve  is  run  in  as  usual. 

If  the  point  of  tangency  is  obstructed,  a  similar  process, 
somewhat  reversed,  may  be  used.  /3  is  that  portion  of  A  still 
to  be  laid  off  when  in  is  reached,  tin  =  il  ==•  It  sin  ft.  mz  = 
tB  =  Ix  =  It  vers  p. 

c.  When  the  central  part  of  the  curve  is  obstructed.  a  is 
the  central  angle  between  two  points  of  the  curve  between  which 
a  chord  may  be  run.  a  may  equal  any  angle,  but  it  is  prefer- 
able that  a  should  be  a  multiple  of  Z>,  the  degree  of  curve,  and 
that  the  points  in  and  n  should  be  on  even  stations,  mn  = 
2 It  sin  \a.  A  point  s  may  be  located 
by  an  offset  ks  from  the  chord  mn  by  a 
similar  method  to  that  outlined  in  §  30. 

The  device  of  introducing  the  dotted 
curve  mn  having  the  same  radius  of  cur- 
vature as  the  other,  although  neither- 
necessary  nor  advisable  in  the  case  shown 
in  Fig.  19,  is  sometimes  the  best  method 
of  surveying  around  an  obstacle.  The 
offset  from  any  point  on  the  dotted  curve 
to  the  corresponding  point  on  the  true  Fm-  19- 

curve  is  twice  the  "ordinate  to  the  long  chord,"  as  computed 
in  §  30. 

33.  Modifications  of  location.  The  following  methods  may 
be  used  in  allowing  for  the  discrepancies  between  the  "  paper 
location  "  based  on  a  more  or  less  rough  preliminary  survey  and 
the  more  accurate  instrumental  location.  (See  §  15.)  They  are 
also  frequently  used  in  locating  new  parallel  tracks  and  modify- 
ing old  tracks. 


RAILROAD  CONSTRUCTION. 


33. 


a.  To  move  the  forward  tangent  parallel  to  itself  a  distance  x> 
the  point  of  curve  (A)  remaining  fixed,     (Fig.  20.) 


Vh  =  B'r  =  a?'. 


FF'  = 


V'h 


sinAFF7  ~sin/7 

A  V  =  A  F+  FF7. 
The  triangle  BmB'  is  isosceles  and  Bm  = 

B'r 


x' 


vers  B'mB      vers 


x 


vers 


(13) 


.     .     .     .     (14) 


The  solution  is  very  similar  in  case  the  tangent  is  moved  in- 
ward to  V" B" .     Note  that  this  method  necessarily  changes  the 


o'  o  o" 


FIG.  20. 


FIG.  21. 


radius.     If  the  radius  is  not  to  be  changed,  the  point  of  curve 
must  be  altered  as  follows : 

b.  To  move  the  forward  tangent  parallel  to  itself  a  distance  x, 
the  radius  being  unchanged.     (Fig.  21.)     In  this  case  the  whole 


§33. 


ALIGNMENT. 


curve  is  moved  bodily  a  distance  00'  =  A  A'  =  VV  = 
and  moved  parallel  to  the  first  tangent  A  V. 


BB'  =  -, 


sin  nBB'    '   sin 


.     (15) 


c.  To  change  the  direction  of  the  forward  tangent  at  the  point 
of  tangency.  (Fig.  22.)  This  problem  involves  a  change  («)  in 
the  central  angle  and  also  requires  a  new  radius.  An  error  in  the 
determination  of  the  central  angle  furnishes  an  occasion  for  its 
use. 

J?,  A,  «,  A  V,  and  B  T^are  known.     A'  =  A  —  OL. 
Bs  =  R  vers  A.         Bs  =  R  vers  A1 '. 
vers  A 


R  =.  R 


vers  (A  —  at)' 
As  =  R  sin  A.  As  =  R  sin  A'. 


(16) 


.-.     AA  =  As  -  As  =  R  sin  A'  -  R  sin  A.  .     (IT) 

The  above  solutions  are  given  to  illustrate  a  large  class  of 
problems  which  are  constantly  arising.     All  of   the  ordinary 


FIG.  22. 


FIG.  23. 


problems  can  be  solved  by  the  application  of  elementary  ge- 
ometry and  trigonometry. 


34  RAILROAD  CONSTRUCTION.  §  34. 

34,  Limitations  in  location,  It  may  be  required  to  run  a 
curve  that  shall  join  two  given  tangents  and  also  pass  through  a 
given  point.  The  point  (jP,  Fig.  23)  is  assumed  to  be  determined 
by  its  distance  ( VP)  from  the  vertex  and  by  the  angle  A  VP 
=  ft. 

It  is  required  to  determine  the  radius  (7?)  and  the  tangent 
distance  (AV).  A  is  known. 


P  VG  =  4(180°  -  J)  -  ft  =  90°  -  (iJ  +  p). 
PP'  =  2VPsmPV&    =2  VP  cos  (£  J  +  ft). 


.-.     SP  = 


sin 


AS  =.  VSP  x  SPr  =  V8P(SP  +  PP'). 


sin  sin 


4-  2  sin  /^  cos  (2^  +  P) 


sn 


sin 
8V 


FP 


sin  /?  sin^  cos  (i^  +  A)].     (18) 


=  AV 


In  the  special  case  in  which  P  is  on  the  median  line  0  V, 
90°  -  J^f,  and  (J  J  +  /?)  =  90°.     Eq.  (18)  then  reduces  to 

VP 

AY=  1      cos        =  VP  cot   J 


as  might  have  been  immediately  derived  from  Eq.  (8). 


§  35.  ALIGNMENT.  35 

Iii  case  the  point  P  is  given  by  the  offset  PK  and  by  the 
distance  F7T,  the  triangle  PJTFmay  be  readily  solved,  giving 
the  distance  VP  and  the  angle  /?,  and  the  remainder  of  the 
solution  will  be  as  above. 

35,  Determination  of  the  curvature  of  existing  track,  (a)  Using 
a  transit.  Set  up  the  transit  at  any  point  in  the  center  of  the 
track.  Measure  in  each  direction  100  feet  to  points  also  in  the 
center  of  the  track.  Sight  on  one  point  with  the  plates  at  0°. 
Plunge  the  telescope  and  sight  at  the  other  point.  The  angle 
between  the  chords  equals  the  degree  of  curvature. 

(b)  Using  a  tape  and  string.  Stretch  a  string  (say  50  feet 
long)  between  two  points  on  the  inside  of  the  head  of  the  outer 
rail.  Measure  the  ordinate  (x)  between  the  middle  of  the  string 
'and  the  head  of  the  rail.  Then 


For,  in  Fig.  24,  since  the  triangles  AGE  and  ADC  are 
similar,  AO  :  AE  :  :  AD  :  DO  or  E  =  \AI?  -=-  x.  When, 
as  is  usual,  the  arc  is  very  short  compared  with 
the  radius,  AD  =  \AB,  very  nearly.  Making 
this  substitution  we  have  Eq.  (19).  "With  a 
chord  of  50  feet  and  a  10°  curve,  the  resulting 
difference  in  x  is  .0025  of  an  inch  —  far  within 
the  possible  accuracy  of  such  a  method.  The 
above  method  gives  the  radius  of  the  inner  head  FlG-  24- 

of  the  outer  rail.  It  should  be  diminished  by  \g  for  the  radius 
of  the  center  of  the  track.  With  easy  curvature,  however,  this 
will  not  affect  the  result  by  more  than  one  or  two  tenths  of  one 
per  cent. 

The  inversion  of  this  formula  gives  the  required  middle  or- 
dinate for  a  rail  on  a  given  curve.  For  example,  the  middle 
ordinate  of  a  30-foot  rail,  bent  for  a  6°  curve,  is 

x  =  900  -T-  (8  X  955)  =  .118  foot  =  1.4  inches. 


36  RAILROAD  CONSTRUCTION.  §  36. 

Another  much  used  rule  is  to  require  the  foreman  to  have  a 
string,  knotted  at  the  centre,  of  such  length  that  the  middle  or- 
dinate,  measured  in  inches,  equals  the  degree  of  curve.  To 
find  that  length,  substitute  (in  eq.  (19))  5730  -r-  D  f or  R  and 
D  -r-  12  for  x.  Solving  for  chord,  we  obtain  chord  =61.8  feet. 
The  rule  is  not  theoretically  exact,  but,  considering  the  uncertain 
stretching  of  the  string,  the  error  is  insignificant.  In  fact,  the 
distance  usually  given  is  62  feet,  which  is  close  enough  for  all 
purposes  for  which  such  a  method  should  be  used. 

36.  Problems.  A  systematic  method  of  setting  down  the 
solution  of  a  problem  simplifies  the  work.  Logarithms  should 
always  be  used,  and  all  the  work  should  be  so  set  down  that  a. 
revision  of  the  work  to  find  a  supposed  error  may  be  readily 
done.  The  value  of  such  systematic  work  will  become  more 
apparent  as  the  problems  become  more  complicated.  The  two/ 
solutions  given  below  will  illustrate  such  work. 

a.  Given  a  3°  curve  beginning  at  Sta.  27  -4-  60  and  running- 
to  Sta.  32  -f-  45.     Compute  the  ordinates   and  offsets  used  in 
locating  the  curve  by  tangential  offsets. 

b.  With  the  same  data  as  above,  compute  the  distances  to 
locate  the  curve  by  offsets  from  the  long  chord. 

c.  Assume  that  in  Fig.  17  ab  is  measured  as  217.6  feet, 
the  angle  ab  V=  17°  42',  and  the  angle  la  V=  21°  14'.     Join 
the  tangents  by  a  4°  30'  curve.     Determine  bl>  and  a  A. 

d.  Assume  that  in  a  case  similar  to  Fig.  18  it  was  noted 
that  a  distance  (As)  equal  to  12  feet  would  clear  the  building. 
Assume  that  A  —  38°  20'  and  that  D  =  4°  40'.     Eequired  the 
value  of  a  and  the  position  of  n.     Solution : 

vers  a  =  As  -f-  E  As  =  12  log  =  1.07918 

E  (for  4°  40'  curve)  log  =  3.08923 

ot=  8°  01'  log  vers  a  =  7.98994 

ns  =  E  sin  a  log  sin  a  =  9.14445 

log  E=  3.08923 
TW  =  171.27  log  -2.23369 


§37. 


ALIGNMENI. 


37 


e.  Assume  that  the  forward  tangent  of  a  3°  20'  curve 
Laving  a  central  angle  of  16°  50'  must  be  moved  3.62  feet 
imcard,  without  altering  the  P.O.  Eequired  the  change  in 
radius. 

f.  Given  two  tangents  making  an  angle  of  36°  18'.  It  is 
required  to  pass  a  curve  through  a  point  93.2  feet  from  the 
vertex,  the  line  from  the  vertex  to  the  point  making  an  angle 
of  4:2°  21'  with  the  tangent.  Required  the  radius  and  tangent 
distance.  Solution:  Applying  eq.  (18),  we  have 


a 

ft  =  42°  21' 

iJ  =  18°  09' 

($4  +  ft)  =  60°  30' 

.20667 

log  sin*  ft  ==  9.65688. .453S2 

2j  9.81987. .......   .66049 

9.9099J .81271 

nat  sin  60°  30'  =  .870J 

1.6836 

VP=    93.2 


tang,  dist.  AY  =  503.56 


=  1536.1 


log=    0.30103 

log  sin  =    9.82844 

log  sin  =    9.49346 

log  cos  =    9.69234 

9.31527 


log=    0.22610 

log=    1.9694i 

2.1955i 

log  sin  \A  =    9.49346 

log=    2.70205 

log  cot  J A  =  10.48437 

3.18642 


1)  =  3°  44' 


COMPOUND    CURVBB. 

37.  Nature  and  use.  Compound  curves  are  formed  by  a 
succession  of  two  or  more  simple  curves  of  different  curvature. 
The  curves  must  have  a  common  tangent  at  the  point  of  com- 
pound curvature  (P.C.C.).  In  mountainous  regions  there  is 
frequently  a  necessity  for  compound  curves  having  several 
changes  of  curvature.  Such  curves  may  be  located  separately 
as  a  succession  of  simple  curves,  but  a  combination  of  two 


38 


RAILROAD  CONSTRUCTION. 


§38. 


simple  curves  has  special  properties  wliicli  are  worth  investigat- 
ing and  utilizing.  In  the  following  demonstrations  7?2  always 
represents  the  longer  radius  and  7^  the  shorter,  no  matter 
which  succeeds  the  other.  Tl  is  the  tangent  adjacent  to  the 
curve  of  shorter  radius. (7?,),  and  is  invariably  the  shorter  tan- 
gent, z/j  is  the  central  angle  of  the  curve  of  radius  72, ,  but  it 
may  be  greater  or  less  than  J2. 

38.  Mutual  relations  of  the  parts  of  a  compound  curve  having 
two  branches,     In  Fig.  25,  AC  and  CB  are  the  two  branches  of 


FIG  25. 

the  compound  curve  having  radii  of  7?,  and  722  and  central 
angles  of  Al  and  J2.  Produce  the  arc  AC  to  n  so  that 
Aojh  =  A.  The  chord  On  produced  must  intersect  B.  The 
line  ns,  parallel  to  OO9 ,  will  intersect  BO^  so  that  Bs  =  sn 
=  0^0l  =  7?,  —  7?,.  Draw  Am  perpendicular  to  O^n.  It  will 
be  parallel  to  Jik. 

Br  =  sn  vers  Bsn  =  (7?2  —  7^,)  vers  J2 ; 

mn  =  A  Ol  vers  A  0,n     —  Rl  vers  A  ; 
Ak  =  A  Fsin  A  Vk        =  T,  sin  A ; 
Ak  =  Jim  =  mn  -\-  nil  =  mn  -\-  Br. 
.\  Tl  sin  A  =  ^  vers  //  +  (7?a  -  J?J  vers  Jt.      .     (20) 


V 

§  38.  ALIGNMENT.  39 

Similarly  it  may  be  shown  that 

T7,  sin  A  —  R^  vers  A  _  (7?3  —  Rt)  vers  Al%     .     (21) 

The  mutual  relations  of  the  elements  of  compound  curves 
may  be  solved  by  these  two  equations.  For  example,  assume 
the  tangents  as  fixed  (A  therefore  known)  and  that  a  curve  of 
given  radius  7?x  shall  start  from  a  given  point  at  a  distance  Tl 
from  the  vertex,  and  that  the  curve  shall  continue  through  a 
given  angle  A^  Eequired  the  other  parts  of  the  curve.  From 
Eq.  (20)  we  have 

'  n        Tt  sin  A  —  El  vers  A 


vers  A 


I  sin  A  —  R.  vers  A 


T9  may  then  be  obtained  from  Eq.  (21). 

As  another  problem,  given  the  location  of  the  two  tangents, 
with  the  two  tangent  distances  (thereby  locating  the  PC  and 
PT\  and  the  central  angle  of  each  curve ;  required  the  two 
radii.  Solving  Eq.  (20)  for  Rl ,  we  have 

T7,  sin  A  —  R^  vers  At 
1  ~       vers  A  —  vers  At 

Similarly  from  Eq.  (21)  we  may  derive 

~  __  T^  sin  A  —  Rt  (vert  A  —  vers  A^ 
vers  ^/, 

Equating  these,  reducing,  and  solving  for  7?, ,  we  have 

T,  sin  A  vers  Al  —  T,  sin  A  (vers  A  —  vers  A^ 
8  ~~  vers  A^  vers  A^  —  (vers  A  —vers  ^,)(vers  A  —  vers  A^'   *     ' 

Although  the  various  elements  may  be  chosen  as  above  with 
considerable  freedom,  .there  are  limitations.  For  example,  in 
Eq.  (22),  since  7?3  is  always  greater  than  Rl ,  the  term  to  be 
added  to  Rl  must  be  essentially  positive — i.e.,  Tl  sin  A  must  be 


40 


RAILROAD  CONSTRUCTION. 


Vers 


greater  than  Rl  vers  A.      This  means  that  T^  >  Rl 

or  that  TI  >  Rl  tan  JJ,  or  that  Tl  is  greater  than  the  corre- 
sponding tangent  on  a  simple  curve.  Similarly  it  may  be 
shown  that  Ta  is  less  than  R^  tan  %A  or  less  than  the  correspond- 
ing tangent  on  a  simple  curve.  Nevertheless  T^  is  always 
greater  than  T^  In  the  limiting  case  when  R^  =  R^  ,  T7,  ==  T 
and  Ja  =  J,. 

39,  Modifications  of  location.  Some  of  these  modifications 
may  be  solved  by  the  methods  used  for  simple  curves.  For 
example  : 

a.  It  is  desired  to  move  the  tangent  VB,  Fig.  26,  parallel  to 
itself  to  V  B'  .  Hun  a  new  curve  from  the  P.  C.  C.  which  shall 
reach  the  new  tangent  at  B  ',  where  the  chord  of  the  old  curve 


FIG.  26. 


FIG.  27. 


intersects  the  new  tangent.    The  solution  is  almost  identical  with 
that  in  §  33,  a. 

b.   Assume  that  it  is  desired  to  change  the  forward  tangent 
(as  above)  but  to  retain  the  same  radius.     In  Fig.  27 

(R,  -  B,)  cos  Ja  =  0,n-, 
(R,  -  R,)  cos  J/  =  O>'. 

s  ^2  —  cos  4'). 


cos  ^/  =  cos  ^2  — 


§39. 


ALIGNMENT. 


41 


The  P.  0.  C.  is  moved  backward  along  the  sharper  curve  an 
angular  distance  of  ^a'  —  ^2  =  A  —  ^/. 

In  case  the  tangent  is  moved  inward  rather  than  outward, 
the  solution  will  apply  by  transposing  ^a  and  J/.  Then  we 
will  have 


(25) 


The  P.  £  (7.  is  then  moved  for- 
ward. 

c.  Assume  the  same  case  as  (b)  ex- 
cept that  the  larger  radius  comes  first 
and  that  the  tangent  adjacent  to  the 
smaller  radius  is  moved.  In  Fig.  28 


-,   cos        = 


x  = 


-  cos 


cos     /  =  cos     x 


.     .     .     .     (26) 


The  P.  C.  (7.  is  moved  forward  along  the  easier  curve  an 
angular  distance  of  ^/  —  ^  =  ^,  —  ^/. 

In  case  the  tangent  is  moved  inward,  transpose  as  before  and 
we  have 


cos        =  cos     l  — 


.     .     .     .     (27) 


The  P.  (7.  (7.  is  moved  'backward. 

d.  Assume  that  the  radius  of  one  curve  is  to  be  altered  with- 
out changing  either  tangent.  Assume  conditions  as  in  Fig.  29. 
For  the  diagrammatic  solution  assume  that  J?a  is  to  be  in- 


RAILROAD  CONSTRUCTION. 


§39. 


creased  by  OtS.  Then,  since  R^  must  pass  through  0,  and  ex- 
tend beyond  O^  a  distance  #,$,  the  locus  of  the  new  center 
must  lie  on  the  arc  drawn  about  0^  as  center  and  with  OS  as 

radius.  The  locus  of  OJ  is  also  given 
by  a  line  O^p  parallel  to  B  V  and  at  a 
distance  of  EJ  (equal  to  8  ...  P.  C.  C.) 
from  it.  The  new  center  is  therefore 
at  the  intersection  #/.  An  arc  witk 
radius  7?/  will  therefore  be  tangent  at 
B'  and  tangent  to  the  old  curve  pro- 
duced at  NEW  P.  C.  C.  Draw  O^n' 
perpendicular  to  O^B.  With  Ot  as 
center  draw  the  arc  0,m,  and  with 
O'  as  center  drawr  the  arc  Ojm'  . 


FIG.  29. 


-        vers 


mB  =  m'B'  = 
//  =  (E^  —  RI)  vers  A^. 


mn  =  m'n'  — 


1.     .     .     .     (28) 


=  O^i'-O.n  =  (EJ-E,)  sin  j/—  (E^ 


sn 


(29) 


This  problem  may  be  further  modified  by  assuming  that  the 
radius  of  the  curve  is  decreased  rather  than  increased,  or  that  the 
smaller  radius  follows  the  larger.  The  solution  is  similar  and 
is  suggested  as  a  profitable  exercise. 

It  might  also  be  assumed  that,  instead  of  making  a  given 
change  in  the  radius  7?,,  a  given  change  BB'  is  to  be  made. 
^/  and  ./?,'  are  required.  Eliminate  7?/  from  Eqs.  28  and  29 
and  solve  the  resulting  equation  for  4  '.  Then  determine  Rj  by 
a  suitable  inversion  of  either  Eq.  28  or  29. 


§  41.  ALIGNMENT.  43 

As  in  §§  32  and  33,  the  above  problems  are  but  a  few, 
although  perhaps  the  most  common,  of  the  problems  the 
engineer  may  meet  with  in  compound  curves.  All  of  the 
ordinary  problems  may  be  solved  by  these  and  similar 
methods. 

40.  Problems,  a.  Assume  that  the  two  tangents  of  a  com- 
pound curve  are  to  be  348  feet  and  624  feet,  and  that  ^l  = 
22°  16'  and  ^a  =  28°  20'.  Kequired  the  radii. 

[Ans.  7^  =  326.92;  Et  =  1574.85.] 

b.  A  line  crosses  a  valley  by  a  compound  curve  which  is  first 
a  6°  curve  for  46°  30'  and  then  a  9°  30'  curve  for  84°  16'.  It  is 
afterward  decided  that  the  last  tangent  should  be  6  feet  farther  up 
the  hill.  "What  are  the  required  changes  ?  {Note.  The  second 
tangent  is  evidently  moved  outward.  The  solution  corresponds 
to  that  in  the  first  part  of  §  39,  c.  The  P.  C.  C.  is  moved  forward 
16.39  feet.  If  it  is  desired  to  know  how  far  the  P.T.  is  moved 
in  the  direction  of  the  tangent  (i.e.,  the  projection  of  BB,  Fig. 
28,  on  VB'),  it  may  be  found  by  observing  that  it  is  equal  to 
nn'  =  (R^  —  7?,)(sin  A^  —  sin  Af).  In  this  case  it  equals  0.65 
foot,  which  is  very  small  because  J,  is  nearly  90°.  The  value 
of  Ja  (46°  SO')  is  not  used,  since  the  solution  is  independent  of 
the  value  of  A^.  The  student  should  learn  to  recognize  which 
quantities  are  mutually  related  and  therefore  essential  to  a  solu- 
tion, and  which  are  independent  and  non-essential.] 


TRANSITION    CURVES. 

41.  Superelevation  of  the  outer  rail  on  curves.  When  a  mass 
is  moved  in  a  circular  path  it  requires  a  centripetal  force  to  keep 
it  moving  in  that  path.  By  the  principles  of  mechanics  we 
know  that  this  force  equals  6V  -r-  gE,  in  which  G  is  the  weight, 
v  the  velocity  in  feet  per  second,  g  the  acceleration  of  gravity 
in  feet  per  second  in  a  second,  and  E  the  radius  of  curvature. 
If  the  two  rails  of  a  curved  track  were  laid  on  a  level  (trans- 
versely), this  centripetal  force  could  only  be  furnished  by  the 


44  RAILROAD  CONSTRUCTION.  §  41. 

pressure  of  the  wheel-flanges  against  the  rails.  As  this  is  very 
objectionable,  the  outer  rail  is  elevated  so  that  the  reaction  of 
the  rails  against  the  wheels  shall  contain 
a  horizontal  component  equal  to  the  re- 
quired centripetal  force.  In  Fig.  30,  if 
ob  represents  the  reaction,  oc  will  repre- 
sent the  weight  6r,  and  ao  will  represent 
the  required  centripetal  force.  From 
similar  triangles  we  may  write  sn :  sm  :  : 
ao  :  oc.  Call  g  =  32.17.  Call  R  = 
5730  ^  7>,  whigh  is  sufficiently  accurate 

for  this  purpose  (see  §  19).  Call  v.=  5280  V -±  3600,  in 
whieh  T^is  the  velocity  in  miles  per  hour,  mn  is  the  distance 
between  rail  centers,  which,  for  an  80-lb.  rail  and  standard 
gauge,  is  4.916  feet,  sm  is  slightly  less  than  this.  As  an 
average  value  we  may  call  it  4.900,  which  is  its  exact  value 
when  the  superelevation  is  4f  inches.  Calling  sn  =  e,  we  have 


ao  —  A  q^'  *          4.9  X  5280'  V*D 
moc~        ~^RG  =  32.17  X  3600s  X  5730* 

e=  .0000572  V*D (30) 


It  should  be  noticed  that,  according  to  this  formula,  the 
required  superelevation  varies  as  the  square  of  the  velocity, 
which  means  that  a  change  of  velocity  of  only  10$  would  call 
for  a  change  of  superelevation  of  21$.  Since  the  velocities  of 
trains  over  any  road  are  extremely  variable,  it  is  impossible  to 
adopt  any  superelevation  which  will  fit  all  velocities  even 
approximately.  The  above  fact  also  shows  why  any  over- 
refinement  in  the  calculations  is  useless  and  why  the  above 
approximations,  which  are  really  small,  are  amply  justifiable. 
For  example,  the  above  formula  contains  the  approximation  that 
R  =  5730  -T-  D.  In  the  extreme  case  of  a  10°  curve  the  error 
involved  would  be  about  1$.  A  change  of  about  i  of  1$  in 


§42. 


ALIGNMENT. 


45 


the  velocity,  or  say  from  40  to  40.2  miles  per  hour,  would  mean 
as  much.  The  error  in  e  due  to  the  assumed  constant  value 
of  sin,  is  never  more  than  a  very  small  fraction  of  \%.  The 
rail-laying  is  not  done  closer  than  this.  The  following  tabular 
form  is  based  on  Eq.  30 : 

SUPERELEVATION  OF  THE  OUTER  RAIL  (IN  FEET)  FOR  VARIOUS  VELOCI- 
TIES AND  DEGREES  OF  CURVATURE. 


Velocity 
iu 
Miles 
pei- 
Huur. 

Degree  of  Curve. 

1° 

2° 

3° 

4° 

5° 

6° 

70 

8° 

9° 

10° 

30 
40 
50 

60 

.05 
.09 
.14 
.20 

.10 
.18 
.29 
.41 

.15 
.27 
.43 

f""6T 

.20 

.37 

.26 
.46 

.31 

7755" 

.86 

.36 

-eT 

.41 
—  7T 

.46 

TsT 

1.51 

.57 
.82 

.71 

42.  Practical  rules  for  superelevation.  A  much  used  rule 
for  superelevation  is  to  "  elevate  one  half  an  inch  for  each 
degree  of  curvature."  The  rule  is  rational  in  that  e  in  Eq.  30 
varies  directly  as  D.  The  above  rule  therefore  agrees  with 
Eq.  30  when  Vis  about  27  miles  per  hour.  However  applica- 
ble the  rule  may  have  been  in  the  days  of  low  velocities,  the 
elevation  thus  computed  is  too  small  now. 

Another  (and  better)  rule  is  to  "  elevate  for  the  speed  of  the 
fastest  trains."  This  rule  is  further  justified  by  the  fact  that  a 
four-wheeled  truck,  having  two  parallel  axles,  will  always  tend 
to  run  to  the  outer  rail  and  will  require  considerable  flange 
pressure  to  guide  it  along  the  curve.  The  effect  of  an  excess  of 
superelevation  on  the  slower  trains  will  only  be  to  relieve  this 
flange  pressure  somewhat.  This  rule  is  coupled  with  the  limita- 
tion that  the  elevation  should  never  exceed  a  limit  of  six  inches 
— sometimes  eight  inches.  This  limitation  implies  that  locomo- 
tive engineers  must  reduce  the  speed  of  fast  trains  around  sharp 
curves  until  the  speed  does  not  exceed  that  for  which  the  actual 
superelevation  used  is  suitable.  The  heavy  line  in  the  tabular 
form  (§  41)  shows  the  six-inch  limitation. 


46  RAILROAD   CONSTRUCTION.  §  43. 

Some  roads  furnish  their  track  foremen  with  a  list  of  the 
superelevations  to  be  used  on  each  curve  in  their  sections. 
This  method  has  the  advantage  that  each  location  may  be 
separately  studied,  and  the  proper  velocity,  as  affected  by  local 
conditions  (e.g.,  proximity  to  a  stopping-place  for  all  trains), 
may  be  determined  and  applied. 

Another  method  is  to  allow  the  foremen  to  determine  the 
superelevation  for  each  curve  by  a  simple  measurement  taken 
at  the  curve.  The  rule  is  developed  as  follows  :  By  an  inversion 
of  Eq.  19  we  have 

sc  =  chord'1  —  872     .....     (31) 

Putting  x  equal  to  e  in  Eq.  30  and  solving  for  "chord,"  we 
have 


?  =  .0000572 
=  2.621  F". 
chord  =1.6W.     .......     (32) 

To  apply  the  rule,  assume  that  50  miles  per  hour  is  fixed  a& 
the  velocity  from  which  the  superelevation  is  to  be  computed. 
Then  1.62F=  1.62  X  50  =  81  feet,  which  is  the  distance  given 
to  the  trackmen.  Stretch  a  tape  (or  even  a  string)  with  & 
length  of  81  feet  between  two  points  on  the  inside  head  of  the 
outer  rail  or  the  outer  head  of  the  inner  rail.  The  ordinate  at 
the  middle  point  then  equals  the  superelevation.  The  values 
of  this  chord  length  for  varying  velocities  are  given  in  the 
accompanying  tabular  form. 


Velocity  in  miles  per  hour  
Chord  length  in  feet              .  . 

20 
32  4 

25 

40  5 

30 
48  6 

35 

56  7 

40 
64  8 

45 
72  9 

50 
81  0 

55 

89  1 

60 
97  2 

43.  Transition  from  level  to  inclined  track,  On  curves  the 
track  is  inclined  transversely;  on  tangents  it  is  level.  The 
transition  from  one  condition  to  the  other  must  be  made  gradn- 


§  45.  ALIGNMENT.  47 

ally.  If  there  is  no  transition  curve,  there  must  be  either  in- 
clined track  on  the  tangent  or  insufficiently  inclined  track  on  the 
curve  or  both.  Sometimes  the  fujl  superelevation  is  continued 
through  the  total  length  of  the  curve  and  the  "  run- oft' " 
(having  a  length  of  100  to  200  feet)  is  located  entirely  on  the 
tangents  at  each  end.  In  other  practice  it  is  located  partly  on 
the  tangent  and  partly  on  the  curve.  Whatever  the  method, 
the  superelevation  is  correct  at  only  one  point  of  the  run-oft'. 
At  all  other  points  it  is  too  great  or  too  small.  This  (and  other 
causes)  produces  objectionable  lurches  and  resistances  when 
entering  and  leaving  curves.  The  object  of  transition  curves  i& 
to  obviate  these  resistances. 

44.  Fundamental  principle  of  transition  curves.     If  a  curve 
has  variable  curvature,  beginning  at  the  tangent  with  a  curve  of 
infinite  radius,  and  the  curvature  gradually  sharpens  until  it 
equals  the  curvature  of  the  required  simple  curve  and  there 
becomes  tangent  to  it,  the  superelevation  of  such  a  transition 
curve  may  begin  at  zero  at  the  tangent,  gradually  increase  to 
the  required  superelevation  for  the  simple  curve,  and  yet  have 
at  every  point  the  superelevation  required  by  the  curvature  at 
that  point.     Since  in  Eq.  (30)  e  is'  directly  proportional  to  Z>, 
the  required  curve  must  be  one  in  which  the  degree  of  curve 
increases  directly  as  the  distance  along  the  curve.     The  mathe- 
matical development  of  such  a  curve  is  quite  complicated.     It 
has,  however,  been  developed,  and  tables  have  been  computed  for 
its  use,  by  Prof.  C.   L.   Crandall.     The  following  method  has 
the  advantage  of  great  simplicity,  while  its  agreement  with  the 
true  transition  curve  is  as  close  as  need  be,  as  will  be  shown. 

45.  Multiform    compound    curves.      If   the   transition   curve 
commences  with  a  very  flat  curve  and  at  regular  even  chord 
lengths  compounds  into  a  curve  of  sharper  curvature  until  the 
desired  curvature  is  reached,  the  increase  in  curvature  at  each 
chord  point  being  uniform,  it  is  plain  that  such  a  curve  is  a 
close  approximation  to  the  true  spiral,  especially  since  the  rails 
as  laid  will  gradually  change  their  curvature  rather  than  main- 
tain  a  uniform   curvature    throughout  each   chord   length   and 


48  HAILROAD   CONSTRUCTION.  §46. 

then  abruptly  change  the  curvature  at  the  chord  points.  Such 
a  curve,  as  actually  laid,  will  be  a  much  closer  approximation 
to  the  true  curve  than  the  multiform  compound  curve  by  which 
it  is  set  out.  There  will  actually  be  a  gradual  increase  in 
curvature  which  increases  directly  as  the  length  of  the  curve. 

46,  Required  length  of  spiral.     The  required  length  of  spiral 
evidently    depends    on    the     amount   of    superelevation    to    be 
gained,  and  also  depends  somewhat  on  the  speed.     If  the  spiral 
is  laid  off  in  25-foot  chord  lengths,  with  the  first  chord  subtend- 
ing a  1°  curve,  the  second  a  2°  curve,  etc.,  the  fifth  chord  will 
subtend  a  5°  curve,  and  the  increase  from  this  last  chord  to  a 
6°   curve  is   the  same    as    the  uniform  increase  of  curvature 
between  the  chords.     The  same  spiral  extended  would  run  on 
to  a  12°  curve  in  (12  -  1)25  =  275  feet.     The  last  chord  of  a 
spiral  should  have  a  smaller  degree  of  curvature  than  the  simple 
curve  to  which  it  is  joined.     If  the  curves  are  very  sharp,  such 
as  are  used  in  street  work  and  even  in  suburban  trolley  work, 
an  increase  in  degree  of  curvature  of  1°  per  25  feet  will  not  be 
sufficiently  rapid,  as  such  a  rate  would  require  too  long  curves. 
2°,  10°,  or  even  20°  increase  per  25  feet  may  be  necessary,  but 
then  the  chords  should  be  reduced  to  5  feet.      Such  a  rapid  rate 
of  increase  is  justified  by  the  necessary  reduction  in  speed.      On 
the   other  hand,  very  high   speed  will   make  a  lower  rate   of 
increase  desirable,  and  therefore  a  spiral  whose  degree  of  curva- 
ture increases  only  0°  30'  per  25  feet  may  be  used.      Such  a 
spiral  would  require  a  length  of  375  feet  to  run  on  to  an  8° 
curve,  which  is  inconveniently  long,  but  it  might  be  used  to 
run  on  to  a  4°  curve,  where  its  length  would  be  only  175  feet. 
Three  spirals  have  been  developed  in  Table  IY,  each  with  chords 
of  25  feet,  the  rate  of  increase  in  the  degree  of  curvature  being 
0°  30',  1°  and  2°  per  chord.      One  of  these  will  be  suitable  for 
any  curvature  found  on  ordinary  steam-railroads. 

47.  To  find  the  ordinates  of  a  l°-per-25-feet  spiral,     Since  the 
first  chord  subtends  a  1°  curve,  its  central  angle  is  0°  15'  and 
the  angle  aQ  V  (Fig.  31)  is  7'  30".     The  tangent  at  a  makes  an 
angle  of  15'  with  VQ.     The  angle  between  the  chord  la  and 


§  48. 


ALIGNMENT. 


49 


the  tangent  at  a  is  £(30')  =  15'> and  tlie  angle  &«&"=  4(30')  +  15' 
=  30'!  Similarly  the  angle  cfo"  =  4(45')  +  30'  +  15'  =  67'  30'' 
=  1°  07'  30",  and  the  angle  dcd"  is  2°  0'.  The  ordinate  aa' 
=  25  sin  7'  30",  and  <)a'  =  25  cos  T  30".  <#'  =  #a'  +  «'&' 
=  Qa  +  ab"  =  25  (cos  7'  30"  +  cos  30').  W  =  Vb"  +  W 
—  25  (sin  7'  30"  +  sin  30').  Similarly  the  ordinates  of  c,  d, 
etc. ,  may  be  obtained. 


FIG.  31. 


Fre.  32. 


48.  To   find  the   deflections    from   any  point   of  the   spiral, 

vQ  V  =  T  30".  Tan  IQ  V  =  W  -5-  QV  •  tan  cQ  V  —  cc'  -5-  Qc' ; 
etc.  Thus  we  are  enabled  to  find  the  deflection  angles  from 
the  tangent  at  Q  to  any  point  of  the  spiral. 

The  tangent  to  the  curve  at  c  (Fig.  32)  makes  an  angle  of 
1°  30'  with  QV,orcmF=l°  30'.    Qcm  =  cm V -  cQm.    The 


RAILROAD   CONSTRUCTION. 


§48. 


value  of  cQm  is  known  from  previous  work.     The  deflection 
from  c  to  Q  then  becomes  known. 

acm  —  cmV—  cap  =  cm V  —  caq  —  qap.  caq  is  the  deflec- 
tion angle  to  c  from  the  tangent  at  a  and  will  have  been 
previously  computed  numerically,  qap  =  15'.  acm  therefore 
becomes  known. 

Icm  =  i  of  45'  =  22'  30"; 

den  =  £  of  60'  =  30'. 

ecn-  ecd11—  ncd",  ncd"  =  cm  V,  tan  ecd"  =  (>?'-  d"d')+  c'e', 
all  of  which  are  known  from  the  previous  work. 

By  this  method  the  deflections  from  the  tangent  at  any 


O' 


FIG.  33. 

point  of  the  curve  to  any  other  point  are  determinable.  These 
values  are  compiled  in  Table  IY.  The  corresponding  values 
of  these  angles  when  the  increase  in  the  degree  of  curvature  per 
chord  length  is  30',  and  when  it  is  2°,  are  also  given  in 
Table  IY. 


§  49.  ALIGNMENT.  51 

49.  Connection  of  spiral  with  circular  curve  and  with  tangent. 
See  Fig.  33.*  Let  A  V  and  BV\>Q  the  tangents  to  be  connected 
by  a  D°  curve,  having  a  suitable  spiral  at  each  end.  If  no 
spirals  were  to  be  used,  the  problem  would  be  solved  as  in 
simple  curves  giving  the  curve  AMB.  Introducing  the  spiral 
has  the  effect  of  throwing  the  curve  away  from  the  vertex  a 
distance  MM'  and  reducing  the  central  angle  of  the  D°  curve 
by  20.  Continuing,  the  curve  beyond  Z  and  Z'  to  A'  and  B'  ', 
we  will  have  A  A  =  BB'  —  MM'.  ZK  '=  the  x  ordinate  and 
is  therefore  known.  Call  MM  '  =  m.  A  'N  =  x  —  It  vers  0. 
Then 

.  (33) 


cos  cos 

NA  —  AA  '  sin  \A  —  (x  —  E  vers  0)  tan  \A. 
VQ  =  QK-  KN+NA  +  A  V 

=  y  —  R  sin  0  +  (x  —  R  vers  0)  tan  \A-\-R  tan  \A 
—  y  —  R  sin  0  -f-  x  tan  £J  -f  JK  cos  0  tan  f4.    .     (34) 

When  ^  'JV  has  already  been  computed,  it  may  be  more  con- 
venient to  write 

VQ  —  y  _j_  E  (tan  JJ  -  sin  0)  +  ^'^  tan  f4.       (35) 


=  E  exsec  ^  +  -          -  (36) 

'   cos    d        cos    ^ 


=  y  —  R  sin  0  +  (a?  —  7?  vers  0)  tan  £J.        (37) 

Example.    To  join  two  tangents  making  an  angle  of  34°  20' 
by  a   5°  40'  curve   and    suitable    spirals.     Use    l°-per-25-feet 

*  The  student  should  at  once  appreciate  the  fact  of  the  necessary  distor- 
tion of  the  figure.  The  distance  MM'  in  Fig.  33  is  perhaps  100  times  its  real 
proportional  value. 


RAILROAD   CONSTRUCTION. 


§50. 


spirals   with   five    chords.     Then 
=  17°  10',  and  y  =  124.942. 


(Eq.  33) 


=  3°  45',   x  =  2.999, 


X  = 

A'N  = 

E 

vers  0 
2.166 
2.999 

3.00497 
7.33063 

0.33560 

9.92064 
9.98021 

0.833 
cos  \A 

m  =  MM'  =  AA'  =     0.872 


(Eq.  36) 


FJf '  =  48036 


(Eq.  35)      y  =  124.942 


246.314 


[See  above] 


(Eq.  37) 


0.257 
VQ  =  371.513 


312.471 
AQ  =     59.042 


AN 
tan    J 


tan  $ 
AV 


9.94043 


E 

3.00497 

exsec  ^d 

8.66863 

VM  =  47.164 

1.67366 

m  =     0.872 

nat. 
nat. 

tan 
sin 

V  = 
0  = 

.30891 
.06540 

9. 
3. 

3865i 
00497 

.24351 
R 

2.39148 
9.92064 
9.48984 
9.41048 

3.00497 

9.48984 
2.49481 


50.  Field-work.  When  the  spiral  is  designed  during  the 
original  location,  the  tangent  distance  VQ  should  be  computed 
and  the  point  Q  located.  It  is  hardly  necessary  to  locate  all  of 
the  points  of  the  spiral  until  the  track  is  to  be  laid.  The 
extremities  should  be  located,  and  as  there  will  usually  be  one 
and  perhaps  two  full  station  points  on  the  spiral,  these  should 


§  61.  ALIGNMENT.  53 

also  be  located.  Z  may  be  located  by  setting  off  QK=  y  and 
KZ  =  x,  or  else  by  the  tabular  deflection  for-  Z  from  Q  and  the 
distance  ZQ,  which  is  the  long  chord.  Setting  up  the  instru- 
ment at  Z  and  sighting  back  at  Q  with  the  proper  deflection,  the 
tangent  at  Z  may  be  found  and  the  circular  curve  located  as 
usual,  its  central  angle  being  ^  —  20.  A  similar  operation  will 
locate  Q'  from  Z1 '. 

To  locate  points  on  the  spiral.  Set  up  at  Q,  with  the  plates 
reading  0°  when  the  telescope  sights  along  VQ.  Set  off  from 
Q  the  deflections  given  in  Table  IY  for  the  instrument  at  Q, 
using  a  chord  length  of  25  feet,  the  process  being  like  the 
method  for  simple  curves  except  that  the  deflections  are  irregu- 
lar. If  a  full  station-point  occurs  within  the  spiral,  interpolate 
between  the  deflections  for  the  adjacent  spiral-points.  For  ex- 
ample, a  spiral  begins  at  Sta.  56  +  15.  Sta.  57  comes  10  feet 
beyond  the  third  spiral  point.  The  deflection  for  the  third  point 
is  35'  0";  for  the  fourth  it  is  56'  15".  £J  of  the  difference 
(21'  15")  is  8'  30"  ;  the  deflection  for  Sta.  57  is  therefore  43'  30". 
This  method  is  not  theoretically  accurate,  but  the  error  is  small. 
Arriving  at  z,  the  forward  alignment  may  be  obtained  by  sight- 
ing back  at  Q  (or  at  any  other  point)  with  the  given  deflection 
for  that  point  from  the  station  occupied.  Then  when  the  plates 
read  0°  the  telescope  will  be  tangent  to  the  spiral  and  to  the 
succeeding  curve.  All  rear  points  should  be  checked  from  z. 
If  it  is  necessary  to  occupy  an  intermediate  station,  use  the  de- 
flections given  for  that  station,  orienting  as  just  explained  for  z, 
checking  the  back  points  and  locating  all  forward  points  up  to  z 
if  possible. 

After  the  center  curve  has  been  located  and  z'  is  reached,  the 
other  spiral  must  be  located  but  in  reverse  order,  i.e.,  the  sharp 
curvature  of  the  spiral  is  at  z'  and  the  curvature  decreases  toward 

Q'- 

51.  To  replace  a  simple  curve  by  a  curve  with  spirals.  This 
may  be  done  by  the  method  of  §  49,  but  it  involves  shifting  the 
whole  track  a  distance  ra,  which  in  the  given  example  equals 
0.87  foot.  Besides  this  the  track  is  appreciably  shortened, 


RAILROAD  CONSTRUCTION. 


§51. 


which  would  require  rail -cutting.  But  the  track  may  be  kept  at 
practically  the  same  length  and  the  lateral  deviation  from  the 
old  track  may  be  made  very  small  by  slightly  sharpening  the 
curvature  of  the  old  track,  moving  the  new  curve  so  that  it  is 
wholly  or  partially  outside  of  the  old  curve,  the  remainder  of  it 
with  the  spirals  being  inside  of  the  old  curve.  It  is  found  by 
experience  that  a  decrease  in  radius  of  from  \<f>  to  5$  will  answer 


FIG.  34. 

the  purpose.     The  larger  the  central  angle  the  less  the  change. 

The  solution  is  as  indicated  in  Fig.  34. 

O'JSr=E' 


=  E  '  cos  0  sec  %4  -(-  x  sec  J^. 
m  =  MM'  =  MV-M'V 
=  E  exsec  $4  -  (  0  '  V  -  E  ') 

=  E  exsec  \A  —  E'  cos  0  sec  J  A  —  x  sec  $4  -j-  E'.     (38) 
AQ  =  QK-KN+NV-  VA 

=  y—E'  sin  0  +  (E1  cos  0  +  x)  tan  JJ  —  E  tan  \A 
=y—E'  sin  0  +  Er  cos  0  tan  \A  —  (E  —  x)  tan  JJ.   (39) 


§  51.  ALIGNMENT.  55 


The  length  of  the  old  curve  from  Q  to  Q'  =  2AQ  +  100-— 

The  length  of  the  new  curve  from  Q  to  Q'  =  2Z  +  100 — ~       , 

in  which  L  is  the  length  of  each  spiral. 

Example.  Suppose  the  old  curve  is  a  7°  30'  curve  with  a 
central  angle  of  38°  40'.  As  a  trial,  compute  the  relative  length 
of  a  new  8°  curve  with  spirals  of  seven  chords.  0  =  7°  0' ; 
\A  =  19°  20' ;  R  (for  the  7°  30'  curve)  =  764.489 ;  R '  (for  the 
8°  curve)  =  716.779 ;  x  =  7.628. 

[Eq.  38]  R  2.88337 

exsec  \A  8.77642 

45.687  1.65979 

R'  =  716.779  ===== 

R'  2.85538 

762.466                                   cos  0  9.99675 

sec  \A  0.02521 


[Eq.  39] 


424.328   352.896 
352.896 


•53.953  2.87734 


0.88241 
0.02521 


8.084 
762.037        762.037 
m  =     0.429 

0.90762 

2.85538 
9.08589 

1.94128 

y  =  174.722                                    R' 
sin  <f> 

87.353 
R' 

COS  (f> 

tan  \A 

249.606 
7?  —  7fi4  4RQ 

2.8553§ 
9.99675 
9.54512 

2.3972g 

x  =      7.628 

756.861 
tan  4^/ 

2.8790i 
9.54512 

265.543  2.4241s 


=  71.432 


56  RAILROAD  CONSTRUCTION.  §  52. 

The  length  of  the  old  curve  from  Q  to  Q'  is 

100^  =  1003-|^-7  =  515.556 

24^  =  2x71.432  =  142.864 

New  curve  :  100^  =  IW?^  ~^  =  308.333        658'42° 
2L  =  2  X  175  =350.000 

658.333        658.333 
Difference  in  length  =      0.087 

Considering  that  this  difference  may  be  divided  among  22 
joints  (using  30-foot  rails)  no  rail-cutting  would  be  necessary. 
If  the  difference  is  too  large,  a  slight  variation  in  the  value  of 
the  new  radius  Rf  will  reduce  the  difference  as  much  as  neces- 
sary. A  truer  comparison  of  the  lengths  would  be  found  by 
comparing  the  lengths  of  the  arcs. 

52.  Application  of  transition  curves  to  compound  curves, 
Since  compound  curves  are  only  employed  when  the  location  is 
limited  by  local  conditions,  the  elements  of  the  compound  curve 
should  be  determined  (as  in  §§  38  and  39)  regardless  of  the 
transition  curves,  depending  on  the  fact  that  the  lateral  shifting 
of  the  curve  when  transition  curves  are  introduced  is  very 
small.  If  the  limitations  are  very  close,  an  estimated  allowance 
may  be  made  for  them. 

Methods  have  been  devised  for  inserting  transition  curves 
between  the  branches  of  a  compound  curve,  but  the  device  is 
complicated  and  usually  needless,  since  when  the  train  is  once  on 
a  curve  the  wheels  press  against  the  outer  rail  steadily  and  a 
change  in  curvature  will  not  produce  a  serious  jar  even  though 
the  superelevation  is  temporarily  a  little  more  or  less  than  it 
should  be. 

If  the  easier  curve  of  the  compound  curve  is  less  than  3°  or 
4°,  there  may  be  no  need  for  a  transition  curve  off  from  that 
branch.  This  problem  then  has  two  cases  according  as  transition 
curves  are  used  at  both  ends  or  at  one  end  only. 


ALIGNMENT. 


57 


a.  With  transition  curves  at  both  ends.  Adopting  the 
method  of  §  49,  calling  ^  =  -J^f,  we  may  compute  ra,  =  JO/j'. 
Similarly,  calling  <4,  =  i^,  we  may  compute  mt  =  MM*.  But 


FIG.  35. 


Jf/  and  Jf/  must  be  made  to  coincide.     This  may  be  done  by 
moving  the  curve  Z'M^  and  its  transition  curve  parallel  to  Q'  V 
a  distance  M^M^  and  the  other  curve  parallel  to  Q  V  a  distance 
J/,' J/,.     In  the  triangle  Jf/J^Jf/,  the  angle  at  M,'=  90°  —  J1? 
the  angle  at  Jf/  =  90°  —  Js,  and  the  angle  at  Jfs  =  ^. 


Then 


Similarly  Jf.'Jf.  =  M^ 
J  *      " 


sin 


sm 


(40) 


58  EAILROAD  CONSTRUCTION.  §53. 

b.  With  a  transition  curve  on  the  sharper  curve  only.  Com- 
pute m,  =  JOf/  as  before ;  then  move  the  curve  Z^MJ  parallel 
to  Q'  V  a  distance  of 


sin 


(41) 
v     ; 


The  simple  curve  JO  is  moved  parallel  to  VA  a  distance  of 


If  ^  and  49  are  both  small,  Jf/J^4  and  JOf4  may  be  more 
than  m,,  but  the  lateral  deviation  of  the  new  curve  from  the  old 
will  always  be  less  than  mlt 

53,  To  replace  a  compound  curve  by  a  curve  with  spirals, 
The  solution  is  somewhat  analogous  to  that  of  §  51.  Compute 
m,  for  the  sharper  branch  of  the  curve,  placing  4l  =  ^A  in  Eq. 
38.  Since  m,  and  w3  for  the  two  branches  of  the  curve  must 
be  identical,  a  value  for  J?/  must  be  found  which  will  satisfy 
the  determined  value  of  ma  =  mt.  Solving  Eq.  38  for  R,  we 
obtain 

4-m  cos        -  a? 


COS  0  —  COS  i^ 

Substituting  in  this  equation  the  known  value  of  ml  (=  m0) 
and  calling  R'  =  RJ,  It  =  -Z?2,  and  ^2  =  \A,  solve  for  RJ. 
Obtain  the  value  of  A  Q  for  each  branch  of  the  curve  separately 
by  Eq.  39,  and  compare  the  lengths  of  the  old  and  new  lines. 

Example.  Assume  a  compound  curve  with./),  =  8° ;  Z>2  =  4° ; 
4  =  36°  and  4,  =  32°.  Use  l°-per-25-feet  spirals ;  0,  =  7°  0' ; 
0a  —  1°  30r.  Assume  that  the  sharper  curve  is  sharpened  from 
8°  (X  to  8°  12'. 


1 63. 

[Eq.  38] 


ALIGNMENT. 


59 


[Eq.  43] 


[Eq.  39] 


169.209 
699.326 

868.535 


867.399 
=      1.136 


217.700 


1424.54 
174.722 


504.302 


679. 024 

600.461 

=    78.563 


exsec  36° 


857.970 


9.429 
867.399 


COS 

sec 


sec 


vers  32° 


m,  =  1.136 
cos  32° 


0.963 
0.763 
1.726 


nat.  cos  0   =  .99966 
nat.  cos  /J3  =  .  84805 


85.226 


tan 


.15161 


sin 


COS  0, 

=  36°] 


=  716.779 
=      7.628 


709.151 


515.235 
600.461 


2.85538 
9.37303 

2.22842 

2.84468 
9.99675 
0.09204 

2.9334? 

0.88241 
0.09^04 

0.97445 


3.15615 
9.18175 


0.05538 
9.92842 

9.98380 


2.33440 

9.18073 
3.15367 

2.84468 
9.08589 

1.93057 

2.84468 
9.99675 
9.86126 

2.70269 


2.85074 
9.86126 

2.71206 


60 

[Eq.30] 


RAILROAD  CONSTRUCTION. 
=    74.994  Sin202 


37  290 


tan 


COS  02 

=  32°) 


389.843 


It*  =  1432.69 
««  =   0.76 

1431.93 


894.770 


964.837 
932.060 


932.060 
=    32.777 
For  the  length  of  the  old  track  we  have : 

100  £-  =  100  f^  =  450. 

J->\ 

100  ^-  =  100  ^  =  800. 

JL/O  4 

AQi  =    78.563 
AQt  =    32.777 

1361.340 
For  the  length  of  the  new  track  we  have  : 


53. 


3.15367 
8.41792 

1.5715Q 

3.15367 
9.99985 
9.79579 

2.9493i 


3.15592 
9.79579 

2.95171 


IV  4°.023 

Spiral  on  8°  12'      curve 
"       "  4°  01'  22"    " 


=    758.140 


175.000 
75. 


Length  of  new  track          =  1361.799 
"  old      "  =  1361.340 

Excess  in  length  of  new  track  =        0,459  feet. 


§  55.  ALIGNMENT.  61 

Since  the  new  track  is  slightly  longer  than  the  old,  it  shows 
that  the  new  track  runs  too  far  outside  the  old  track  at  the 
P.  C.  C.  On  the  other  hand  the  offset  m  is  only  1.136.  The 
maximum  amount  by  which  the  new  track  comes  inside  of  the 
old  track  at  two  points,  presumably  not  far  from  Z1  and  Z,  is 
very  difficult  to  determine  exactly.  Since  it  is  desirable  that  the 
maximum  offsets  (inside  and  outside)  should  be  made  as  nearly 
equal  as  possible,  this  feature  should  not  be  sacrificed  to  an  effort 
to  make  the  two  lines  of  precisely  equal  length  so  that  the  rails 
need  not  be  cut.  Therefore,  if  it  is  found  that  the  offsets  inside 
the  old  track  are  nearly  equal  to  m  (1.136),  the  above  figures 
should  stand.  Otherwise  m  may  be  diminished  (and  the  above 
excess  in  length  of  track  diminished)  by  increasing  RJ  very 
slightly  and  making  the  necessary  consequent  changes. 


VERTICAL    CURVES. 

54.  Necessity  for  their  use.     Whenever  there  is  a  change  in 
the  rate  of  grade,  it  is  necessary  to  eliminate  the  angle  that 
would  be  formed  at  the  point  of  change  and  to  connect  the  two 
grades  by  a  curve.     This  is  especially  necessary  at  a  sag  be- 
tween two  grades,  since  the  shock  caused  by  abruptly  forcing 
an  upward  motion  to  a  rapidly  moving   heavy  train  is   very 
severe  both  to  the  track  and  to  the  rolling  stock. 

55.  Required  length.     Theoretically  the  length  should  de- 
pend on  the  change  in  the  rate  of   grade,  the  greater  change 
requiring  a  longer  curve.     The  importance  of  this  was  greater 
in  the  days  when  link  couplers  were  in  universal  use  and  the 
"  slack  "  in  a  long  train  was  very  great.     Under  such  circum- 
stances, when  a  train  was  moving  down  a  heavy  grade  the  cars 
would  crowd  ahead  against  the  engine.     Reaching  the  sag,  the 
engine  would  begin  to  pull  out,  rapidly  taking  out  the  slack. 
Six  inches  of  slack  on  each  car  would  amount  to  several  feet  on 
a  long  train,  and  the  resulting  jerk  on  the  couplers,  especially 
those   near  the  rear  of  the  train,  has  frequently  resulted  in 


62  RAILROAD  CONSTRUCTION.  §  56. 

broken  couplers  or  even  derailments.  A  vertical  curve  will 
practically  eliminate  this  danger  if  the  curve  is  made  long 
enough,  but  the  rapidly  increasing  adoption  of  close  spring 
couplers  and  air-brakes,  even  for  freight  trains,  is  obviating  the 
necessity  for  such  very  long  curves.  Two  hundred  feet  may  be 
considered  sufficiently  long  for  all  ordinary  changes  of  grade. 
Four  hundred  feet  would  probably  suffice  for  the  greatest 
change  ever  found  in  practice. 

56.  Form  of  curve.     In  Fig.  36  assume  that  A  and  (7,  equi- 


FIG.  36. 

distant  from  B,  are  the  extremities  of  the  vertical  curve.  Bi- 
sect AC  at  0;  draw  Be  and  bisect  it  at  h.  Bisect  AB  and  BC 
at  Jc  and  I.  The  line  Id  will  pass  through  h.  A  parabola  may 
be  drawn  with  its  vertex  at  h  which  will  be  tangent  to  AB  and 
BC  at  A  and  B.  It  may  readily  be  shown  from  the  proper- 
ties of  a  parabola  that  if  an  ordinate  be  drawn  at  any  point  (as 
at  n)  we  will  have 

sn :  eh  (or  h,B] : :  Am  :  Ae ', 
or  sn  =  eh—j-^- (44) 


Since  the  elevation  of  any  point  along  AB  or  j^^is  readily 
determinable,  the  elevation  of  any  point  on  the  curve  may  be 
computed  by  adding  the  correction  sn. 

57.  Numerical  example.  Assume  that  B  is  located  at  Sta. 
16  +  20 ;  that  the  curve  is  to  be  200  feet  long;  that  the  grade 
of  AB  is  —  0.8#,  and  of  BC '  +  1.2$;  also  that  the  elevation 
of  B  above  the  datum  plane  is  162.6.  Then  the  elevation  of 
the  various  points  is  as  follows:  A,  163.4;  C,  163.8;  0, 


§  57.  ALIGNMENT.  6S 

|(163. 4  +  163. 8)  =  163. 6;  A,  £(163.6+162.6)  =  163.1.     Then 
eh  =  0.5.     The  elevations  of  the  points  on  the  curve  are: 

Sta.  15  +  20,  (A)  163.4 

"    16  ,  163.4- (.80  x  0.8)  + (.803X  0.5)  =  163. OS 

"    IT  ,  162.6  +  (.80  X  1.2)  +  (.20'  X  0.5)  =  163.58 

"    17  +  20,  (C)  163.8 

A  theoretical  inaccuracy  in  the  above  method  lies  in  the  fact 
that  eh  and  all  parallel  lines  are  not  truly  vertical.  In  the 
above  case  the  variation  from  the  vertical  is  0°  07',  while  the 
effect  of  this  variation  on  the  elevations  in  this  case  (as  in  the 
most  extreme  cases)  is  absolutely  inappreciable.  The  grades 
in  the  figure  are  necessarily  very  greatly  exaggerated,  which 
increases  the  apparent  inaccuracy. 


CHAPTER  III. 
EARTHWORK. 

FORM    OF    EXCAVATIONS    AVD    EMBANKMENTS. 

58.  Usual  form  of  cross-section  in  cut  or  fill.  The  normal 
iorm  of  cross-section  in  cut  is  as  shown  in  Fig.  37,  in  which 
e  .  .  .  g  represents  the  natural  surface  of  the  ground,  no  matter 
how  irregular ;  db  represents  the  position  and  width  of  the  re- 


quired roadbed ;  ac  and  ~bd  represent  the  f  c  side  slopes  ' '  which 
begin  at  a  and  5  and  which  intersect  the  natural  surface  at  such 


FIG.  38. 


points  (G  and  d)  as  will  be  determined  by  the  required  slope 
angle  (ft). 


64 


g  60.  EARTHWORK.  65 

The  normal  section  in  fill  is  as  shown  in  Fig.  38.  The  points 
c  and  d  are  likewise  determined  by  the  intersection  of  the  re- 
quired side  slopes  with  the  natural  surface.  In  case  the  required 
roadbed  (ab  in  Fig.  39)  intersects  the  natural  surface,  both  cut 


FIG.  39. 

and  fill  are  required,  and  the  points  c  and  d  are  determined  as 
before.  Note  that  ft  and  ft'  are  not  necessarily  equal.  Their 
proper  values  will  be  discussed  later. 

59.  Terminal  pyramids  and  wedges.     Fig.  40  illustrates  the 
general  form  of  cross- sections  when  there  is  a  transition  from 
cut  to  fill,     a  .  .  .  g  represents  the  grade  line  of  the  road  which 
passes  from  cut  to  fill  at  d.     sdt  represents  the  surface  profile. 
A  cross-section  taken  at  the  point  where  either  side  of  the  road- 
bed first  cuts  the  surface  (the  point  m  in  this  case)  will  usually 
be  triangular  if  the  ground  is  regular.     A  similar  cross-section 
should  be  taken  at  0,  where  the  other  side  of  the  roadbed  cuts 
the  surface.     In  general  the  earthwork  of  cut  and  fill  terminates 
in  two  pyramids.     In  Fig.  40  the  pyramid  vertices  are  at  n 
and  &,  and  the  bases  are  Ihm  and  opq.     The  roadbed  is  generally 
wider  in  cut  than  in  fill,  and  therefore  the  section  Ihm  and  the 
altitude  In  are  generally  greater  than  the  section  opq  and  the 
altitude  pk.     When  the  line  of  intersection  of  the  roadbed  and 
natural  surface  (nodkm)  becomes  perpendicular  to  the  axis  of 
the  roadbed  (ag)  the  pyramids  become  wedges  whose  bases  are 
the  nearest  convenient  cross- sections. 

60.  Slopes,     a.  Cuttings.     The  required  slopes  for  cuttings 
vary  from  perpendicular  cuts,  which  may  be  used  in  hard  rock 
which  will  not  disintegrate  by  exposure,  to  a  slope_of  perhaps 


66 


RAILROAD   CONSTRUCTION. 


§60. 


4  horizontal  to  1  vertical  in  a  soft  material  like  quicksand  or  in 
a  clayey  soil  which  flows  easily  when  saturated.  For  earthy 
materials  a  slope  of  1  :  1  is  the  maximum  allowable,  and  even 
this  should  only  be  used  for  firm  material  not  easily  affected  by 


FIG.  40. 

saturation.  A  slope  of  1|  horizontal  to  1  vertical  is  a  safer 
slope  for  average  earthwork.  It  is  a  frequent  blunder  that 
slopes  in  cuts  are  made  too  steep,  and  it  results  in  excessive  work 
in  clearing  out  from  the  ditches  the  material  that  slides  down, 
at  a  much  higher  cost  per  yard  than  it  would  have  cost  to  take 
it  out  at  first,  to  say  nothing  of  the  danger  of  accidents  from 
possible  landslides. 

b.  Embankments.  The  slopes  of  an  embankment  vary  from 
1 :  1  to  1.5  :  1.  A  rock  fill  will  stand  at  1:1,  and  if  some  care 
is  taken  to  form  the  larger  pieces  on  the  outside  into  a  rough 
dry  wall,  a  much  steeper  slope  can  be  allowed.  This  method  is 
sometimes  a  necessity  in  steep  side-hill  work.  Earthwork  em- 
bankments generally  require  a  slope  of  1J  to  1.  If  made 
steeper  at  first,  it  generally  results  in  the  edges  giving  way,  re- 
quiring repairs  until  the  ultimate  slope  is  nearly  or  quite  1J- :  1. 
The  difficulty  of  incorporating  the  added  material  with  the  old 
embankment  and  preventing  its  sliding  off  frequently  makes 
these  repairs  disproportionately  costly. 


§  62.  EARTHWORK  67 

61.  Compound   sections.      When  the  cut  consists  partly  of 
earth  and  partly  of  rock,   a  compound  cross-section  must   be 


made.  If  borings  have  been  made  so  that  the  contour  of  the 
rock  surface  is  accurately  known,  then  the  true  cross-section  may 
be  determined.  The  rock  and  earth  should  be  calculated  sepa- 
rately, and  this  will  require  an  accurate  knowledge  of  where  the 
rock  "  runs  out  " — a  difficult  matter  when  it  must  be  deter- 
mined by  boring.  During  construction  the  center  part  of  the 
earth  cut  would  be  taken  out  first  and  the  cut  widened  until  a 
sufficient  width  of  rock  surface  had  been  exposed  so  that  the 
rock  cut  would  have  its  proper  width  and  side  slopes.  Then  the 
earth  slopes  could  be  cut  down  at  the  proper  angle.  A  ' '  berm  ' ' 
of  about  three  feet  is  usually  left  on  the  edges  of  the  rock  cut  as 
a  margin  of  safety  against  a  possible  sliding  of  the  earth  slopes. 
After  the  work  is  done,  the  amount  of  excavation  that  has  been 
made  is  readily  computable,  but  accurate  preliminary  estimates 
are  difficult.  The  area  of  the  cross- section  of  earth  in  the  figure 
must  be  determined  by  a  method  similar  to  that  developed  for 
bor row-pits  (see  §  89). 

62.  Width  of  roadbed.  Owing  to  the  large  and  often  dis- 
proportionate addition  to  volume  of  cut  or  fill  caused  by  the  ad- 
dition of  even  one  foot  to  the  width  of  roadbed,  there  is  a 
natural  tendency  to  reduce  the  width  until  embankments  become 
unsafe  and  cuts  are  too  narrow  for  proper  drainage.  The  cost 
of  maintenance  of  roadbed  is  so  largely  dependent  on  the  drain- 
age of  the  roadbed  that  there  is  true  economy  in  making  an 


68 


RAILROAD  CONSTRUCTION. 


63. 


ample  allowance  for  it.  The  practice  of  some  of  the  leading 
railroads  of  the  country  in  this  respect  is  given  in  the  following 
table,  in  which  are  also  given  some  data  belonging  more  properly 
to  the  subject  of  superstructure. 

WIDTH  OP  ROADBED  FOR  SINGLE  AND  DOUBLE  TRACK-SLOPE  RATIOS- 
DISTANCES  BETWEEN  TRACK  CENTERS. 


Road. 

Single  Track. 

Double  Track. 

Slope 
Ratios. 

Dist.  between 
Track  Centers. 

Cut. 

Fill. 

Cut. 

Fill. 

Cut. 

Fill. 

A.,  T.  &  Santa  Fe.... 

Chi.,  Burl.  &  Quincy 
Chi.,  Mil.  &  St.  Paul. 
C.,  C.,  C.  &  St.  Louis 
Illinois  Central..  ... 
Erie  .  .  

j   28'  earth 
1    22'  rock 
14  -j-  (•>  X  5)  * 
18  +  (2  X  6) 
20  +  (2  X  4) 
32.5 
20'  8J^" 
14  +  (2X3.  5) 

20 

1  :  1 

i!s  !  1 

1.5  : 
1.5  : 
1.5: 
1  :  1 
1.5: 
1  :  1 
1.5: 
1.5: 
1.5  : 

1.5:  1 
1  :  1 

1.5:  1 

1.5:  1 
1.5:  1 
1.5:  1 
1.5  :  1 
1.5  :  1 
1.5:  1 
1.5:  1 
1.5:1 
1.5;  1 
1.5:  1 
1.5:  1 

1.5:  1 
1.5  :  1 

14' 
13' 
13' 

13' 
13' 
13' 

13' 
12' 
18'* 

13' 

12'  2" 

16 
20  to  24 
20 

18 

20'  sy2" 

28  +  (2  X  5) 
31  +  (2  X  6) 
33  +  (2  X  4) 

30 

33  to  37 
33 

33'  8^" 
27  +  (2  X3.5) 
33  +  (2X7.25; 

33'  8J4" 
32 

"33"" 
30 
30'  2" 

31'  4" 

Lehigh  Valley  
L.  S.  &  Michigan  So. 
Louisville  &  Nashv.  . 

13  +  (2X4.  5) 

16 

33  +  (2X2.  5) 
30 
34'  2"  earth 

29'       rock 

31'4"+(2X4) 

N.  Y.  N.  H.  &H.... 
Norfolk  &  Western... 

Pennsylvania  -j 
Union  Pacific 

V  21'  2"  earth"' 
"j  16'       rock 

19'  2"  light  traffic 
27'  2"  heavy    " 
14  -HO  X  3.  5) 

"\rz" 

19'  2" 
19'  2" 
16 

*  (2  X  5)  signifies  two  ditches  each  5  feet  wide:  the  following  cases  should  be  interpreted 
similarly. 

It  may  be  noted  from  the  above  table  that  the  average  width 
for  an  earthwork  cut,  single  track,  is  about  24.7  feet,  with  a 
minimum  of  19  feet  2  inches.  The  widths  of  fills,  single  track, 

average    over  18  feet,   with  numerous  minimums  of  16  feet, 
o 

The  widths  for  double  track  may  be  found  by  adding  the  distance 
between  track  centers,  which  is  usually  13  feet. 

63.  Form  of  subgrade.  TJie  stability  of  the  roadbed  depends 
largely  on  preventing  the  ballast  and  subsoil  from  becoming 
saturated  with  water.  The  ballast  must  be  porous  so  that  it 
will  not  retain  water,  and  the  subsoil  must  be  so  constructed  that  it 
will  readily  drain  off  the  rain-water  that  soaks  through  the  ballast. 
This  is  accomplished  by  giving  the  subsoil  a  curved  form,  convex 


§  64.  EARTHWORK.  69 

upward,  or  a  surface  made  up  of  two  or  three  planes,  the  two 
outer  planes  having  a  slope  of  about  1  :  24  (sometimes  more 
and  sometimes  less,  depending  on  the  soil)  and  the  middle  plane, 
if  three  are  used,  being  level.  When  a  circular  form  is  used, 
a  crowning  of  6  inches  in  a  total  width  of  17  or  18  feet  is  gen- 
erally used.  Occasionally  the  subgrade  is  made  level,  especially 
in  rock-cuts,  but  if  the  subsoil  is  previously  compressed  by 
rolling,  as  required  on  the  N.  Y.  C.  &  H.  E.  E.  E.,  or  if  the 
subsoil  is  drained  by  tile  drains  laid  underneath  the  ditches,  the 
necessity  for  slopes  is  not  so  great.  Eock  cuts  are  generally 
required  to  be  excavated  to  one  foot  below  subgrade  and  then 
filled  up  again  to  subgrade  with  the  same  material,  if  it  is  suit- 
able. 

64.  Ditches.  ' '  The  stability  of  the  track  depends  upon  the 
strength  and  permanence  of  the  roadbed  and  structures  upon 
which  it  rests ;  whatever  will  protect  them  from  damage  or  pre- 
vent premature  decay  should  be  carefully  observed.  The  worst, 
enemy  is  WATER,  and  the  further  it  can  be  kept  away  from  the 
track,  or  the  sooner  it  can  be  diverted  from  it,  the  better  the 
track  will  be  protected.  Cold  is  damaging  only  by  reason  of 
the  water  which  it  freezes ;  therefore  the  first  and  most  impor- 
tant provision  for  good  track  is  drainage. ' '  (Eules  of  the  Eoad 
Department,  Illinois  Central  E.  E.) 

The  form  of  ditch  generally  prescribed  has  a  flat  bottom  12" 
to  24"  wide  and  with  sides  having  a  minimum  slope,  except  in 
rock- work,  of  1:1,  more  generally  1.5 :  1  and  sometimes  2:1. 
Sometimes  the  ditches  are  made  Y-shaped,  which  is  objection- 
able unless  the  slopes  are  low.  The  best  form  is  evidently  that 
which  will  cause  the  greatest  flow  for  a  given  slope,  and  this 
will  evidently  be  the  form  in  which  the  ratio  of  area  to  wetted 
perimeter  is  the  largest.  The  semicircle  ful- 
fills this  condition  better  than  any  other 
form,  but  the  nearly  vertical  sides  would  be 
difficult  to  maintain.  (See  Fig.  42.)  A  ditch,  FIG.  42. 

with  a  flat  bottom  and  such  slopes  as  the  soil  requires,  which 
approximates  to  the  circular  form  will  therefore  be  the  best. 


70  RAILROAD   CONSTRUCTION.  §  65. 

When  the  flow  will  probably  be  large  and  at  times  rapid  it 
will  be  advisable  to  pave  the  ditches  with  stone,  especially  if  the 
soil  is  easily  washed  away.  Six-inch  tile  drains,  placed  2'  under 
the  ditches,  are  prescribed  on  some  roads.  (See  Fig.  43.)  ~No 
better  method  could  be  devised  to  insure  a  dry  subsoil.  The 
ditches  through  cuts  should  be  led  off  at  the  end  of  the  cut  so 
that  the  adjacent  embankment  will  not  be  injured. 

Wherever  there  is  danger  that  the  drainage  from  the  land 
above  a  cut  will  drain  down  into  the  cut,  a  ditch  should  be  made 
near  the  edge  of  the  cut  to  intercept  this  drainage,  and  this 
ditch  should  be  continued,  and  paved  if  necessary,  to  a  point 
where  the  outflow  will  be  harmless.  Neglect  of  these  simple 
and  inexpensive  precautions  frequently  causes  the  soil  to  be 
loosened  on  the  shoulders  of  the  slopes  during  the  progress  of  a 
heavy  rain,  and  results  in  a  landslide  which  will  cost  more  to 
repair  than  the  ditches  which  would  have  prevented  it  for  all 
time. 

Ditches  should  be  formed  along  the  bases  of  embankments ; 
they  facilitate  the  drainage  of  water  from  the  embankment,  and 
may  prevent  a  costly  slip  and  disintegration  of  the  embankment. 

65.  Effect  of  sodding  the  slopes,  etc.  Engineers  are  unani- 
mously in  favor  of  rounding  off  the  shoulders  and  toes  of 
embankments  and  slopes,  sodding  the  slopes,  paving  the  ditches, 
and  providing  tile  drains  for  subsurface  drainage,  all  to  be  put 
in  during  original  construction.  (See  Fig.  43.)  Some  of  the 
highest  grade  specifications  call  for  the  removal  of  the  top  layer 
of  vegetable  soil  from  cuts  and  from  under  proposed  fills  to 
some  convenient  place,  from  which  it  may  be  afterwards  spread 
on  the  slopes,  thus  facilitating  the  formation  of  sod  from  grass- 
seed.  But  while  engineers  favor  these  measures  and  their 
economic  value  may  be  readily  demonstrated,  it  is  generally 
impossible  to  obtain  the  authorization  of  such  specifications 
from  railroad  directors  and  promoters.  The  addition  to  the 
original  cost  of  the  roadbed  is  considerable,  but  is  by  no  means 
as  great  as  the  capitalized  value  of  the  extra  cost  of  mainte- 
nance resulting  from  the  usual  practice.  Fig.  43  is  a  copy  of 


§65. 


EARTHWORK. 


71 


C1JSTOMARY  SECTION  °F 


PROPOSED    SECTION    OF    ROADBED    IN    EXCAVATION 


CUSTOMARY    SECTION    OF    ROADBED    ON    EMBANKMENT. 

GRAVED 


PROPOSED  SECTION    OF.  ROADBED    ON    EMBANKMENT. 

If" 


Sc^^^M^M^ 


FIG.  43. — "  WHITTEMORB  ON  RAILWAY  EXCAVATION  AND  EMBANKMENTS/ 
Trans.  Am.  Soc.  C.  E.,  Sept.  1894  , 


72  RAILROAD   CONSTRUCTION.  §  66, 

designs  *  presented  at  a  convention  of  the  American  Society  of 
Civil  Engineers  by  Mr.  D.  J.  Whittemore,  Past  President  of 
the  Society  and  Chief  Engineer  of  the  Chi.,  Mil.  &  St.  Paul 
R.R.  The  "  customary  sections  "  represent  what  is,  with  some 
variations  of  detail,  the  practice  of  many  railroads.  The  "  pro- 
posed sections"  elicited  unanimous  approval.  They  should  be 
adopted  when  not  prohibited  by  financial  considerations. 


EAKTHWOKK    SURVEYS. 

66.  Relation  of  actual  volume  to  the  numerical  result.     It 
should  be  realized  at  the  outset  that  the  accuracy  of  the  result 
of  computations  of  the  volume  of  any  given  mass  of  earthwork 
has  but  little  relation  to  the  accuracy  of  the  mere  numerical 
work.     The  process  of  obtaining  the  volume  consists  of  two 
distinct  parts.     In  the  first  place  it  is  assumed  that  the  volume 
of  the  earthwork  may  be  represented  by  a  more  or  less  com- 
plicated geometrical  form,  and  then,  secondly,  the  volume  of 
such  a  geometrical  form  is  computed.     A  desire  for  simplicity 
(or  a  frank  willingness  to  accept  approximate  results)  will  often 
cause  the  cross-section  men  to  assume  that  the  volume  may  be 
represented  by  a  very  simple  geometrical  form  which  is  really 
only  a  very  rough  approximation  to  the  true  volume.     In  such 
a  case,  it  is  only  a  waste  of  time  to  compute  the  volume  with 
minute   numerical   accuracy.      One  of  the  first  lessons   to  be 
learned  is  that  economy  of  time  and  effort  requires  that  the 
accuracy  of  the  numerical  work  should  be  kept  proportional  to 
the  accuracy  of  the  cross-sectioning  work,   and   also  that  the 
accuracy  of  both  should  be  proportional  to  the  use  to  be  made 
of  the  results.     The  subject  is  discussed  further  in  §  94. 

67.  Prismoids.      To  compute  the  volume  of  earthwork,  it  is 
necessary  to  assume  that  it  has  some  geometric  form  whose  vol- 
ume is  readily  determinable.    The  general  method  is  to  consider 

*  Trans.  Am.  Soc.  Civil  Eng.,  Sept.  1894. 


§  68.  EARTHWORK.  73 

the  volume  as  consisting  of  a  series  of  prismoids^  which  are 
solids  having  parallel  plane  ends  and  bounded  by  surfaces  which 
may  be  formed  by  lines  moving  continuously  along  the  edges  of 
the  bases.  These  surfaces  may  also  be  considered  as  the  sur- 
faces generated  by  lines  moving  along  the  edges  joining  the  cor- 
responding points  of  the  bases,  these  edges  being  the  directrices, 
and  the  lines  being  always  parallel  to  either  base,  which  is  a 
plane  director.  The  surfaces  thus  developed  may  or  may  not 
be  planes.  The  volume  of  such  a  prismoid  is  readily  determi- 
nable  (as  explained  in  §  70  et  seq.\  while  its  definition  is  so  very 
general  that  it  may  be  applied  to  very  rough  ground.  The 
"  two  plane  ends  "  are  sections  perpendicular  to  the  axis  of  the 
road.  The  roadbed  and  side  slopes  (also  plane)  form  three  of 
the  side  surfaces.  The  only  approximation  lies  in  the  degree  of 
accuracy  with  which  the  plane  (or  warped)  surfaces  coincide  with 
the  actual  surface  of  the  ground  between  these  two  sections. 
This  accuracy  will  depend  (a)  on  the  number  of  points  which 
are  taken  in  each  cross- section  and  the  accuracy  with  which  the 
lines  joining  these  points  coincide  with  the  actual  cross-sections ; 
(b)  on  the  skill  shown  in  selecting  places  for  the  cross-sections  so 
that  the  warped  surfaces  shall  coincide  as  nearly  as  possible  with 
the  surface  of  the  ground.  In  fairly  smooth  country,  cross- 
sections  every  100  feet,  placed  at  the  even  stations,  are  suf- 
ficiently accurate,  and  such  a  method  simplifies  the  computations, 
greatly;  but  in  rough  country  cross-sections  must  be  inter- 
polated as  the  surface  demands.  As  will  be  explained  later,, 
carelessness  or  lack  of  judgment  in  cross-sectioning  will  introduce 
errors  of  such  magnitude  that  all  refinements  in  the  computations 
are  utterly  wasted. 

68.  Cross-sectioning.  The  process  of  cross- sectioning  con- 
sists in  determining  at  any  place  the  intersection  by  a  vertical 
plane  of  the  prism  of  earth  lying  between  the  roadbed,  the  side 
slopes,  and  the  natural  surface.  The  intersection  with  the  road- 
bed and  side  slopes  gives  three  straight  lines.  The  intersection 
with  the  natural  surface  is  in  general  an  irregular  line.  On 
smooth  regular  ground  or  when  approximate  results  are  accept- 


74 


RAILROAD   CONSTRUCTION. 


able  this  line  is  assumed  to  be  straight.  According  to  the  irreg- 
ularity of  the  ground  and  the  accuracy  desired  more  and  more 
4 '  intermediate  points  "  are  taken. 

The  distance  (d  in  Fig.  44)  of  the  roadbed  below  (or  above) 
the  natural  surface  at  the  center  is  known  or  determined  from 


FIG.  44. 

the  profile  or  by  the  computed  establishment  of  the  grade  line. 
The  distances  out  from  the  center  of  all ' '  breaks ' '  are  determined 
with  a  tape.  To  determine  the  elevations  for  a  cut,  set  up  a 
level  at  any  convenient  point  so  that  the  line  of  sight  is  higher 
than  any  point  of  the  cross-section,  and  take  a  rod  reading  on 
the  center  point.  This  rod  reading  added  to  d  gives  the  height 
of  the  instrument  (H.  I.)  above  the  roadbed.  Subtracting  from 
H.  I.  the  rod  reading  at  any  "  break  "  gives  the  height  of  that 
point  above  the  roadbed  (h^  kt,  hr,  etc.).  This  is  true  for  all 
cases  in  excavation.  For  fill,  the  rod  reading  at  center  minus 
d  equals  the  H.  I.,  which  may  be  positive  or  negative.  When 
negative,  add  to  the  "  H.  I."  the  rod  readings  of  the  inter- 
mediate points  to  get  their  depths  below  "  grade  "  ;  when  posi- 
tive, subtract  the  "  H.  I."  from  the  rod  readings. 

The  heights  or  depths  of  these  intermediate  points  above  or 
below  grade  need  only  be  taken  to  the  nearest  tenth  of  a  foot, 
and  the  distances  out  from  the  center  will  frequently  be  suffi- 


§  69. 


EARTHWORK. 


75 


ciently  exact  when  taken  to  the  nearest  foot.  The  roughness  of 
the  surface  of  fanning  land  or  woodland  generally  renders  use- 
less any  attempt  to  compute  the  volume  with  any  greater  accu- 
racy than  these  figures  would  imply  unless  the  form  of  the  ridges 
and  hollows  is  especially  well  defined.  The  position  of  the  slope- 
stake  points  is  considered  in  the  next  section.  Additional  dis- 
cussion regarding  cross-sectioning  is  found  in  §  82. 

69.  Position  of  slope-stakes.  The  slope-stakes  are  set  at  the 
intersection  of  the  required  side  slopes  with  the  natural  surface, 
which  depends  on  the  center  cut  or  fill  (d).  The  distance  of 


FIG.  45. 

the  slope-stake  from  the  center  for  the  lower  side  is  x  =  f b 
-f-  s(d  +  y) ;  for  the  up-hill  side  it  is  x  —  \~b  +  s(d  —  y'). 
s  is  the  ' i  slope  ratio  ' '  for  the  side  slopes,  the  ratio  of  horizontal 
to  vertical.  In  the  above  equation  both  x  and  y  are  unknown. 
Therefore  some  position  must  be  found  by  trial  which  will  sat- 
isfy the  equation.  As  a  preliminary,  the  value  of  x  for  the 
point  a  =  JJ  -f-  sd,  which  is  the  value  of  x  for  level  cross- 
sections.  In  the  case  of  fills  on  sloping  ground  the  value  of  x 
on  the  down-hill  side  is  greater  than  this ;  on  the  up-hill  side  it 
is  less.  The  difference  in  distance  is  s  times  the  difference  of 
elevation.  Take  a  numerical  case  corresponding  with  Fig.  45. 
The  rod  reading  on  c  is  2.9  ;  d  —  4.2 ;  therefore  the  telescope  is 
4.2  —  2.9  =  1.3  below  grade.  *  =  1.5  :  1,  5  =  16.  Hence  for 
the  point  a  (or  for  level  ground)  x  =  %  X  16  +  1. 5x4. 2  = 
14.3.  At  a  distance  out  of  14.3  the  ground  is  seen  to  be  about  3 
feet  lower,  which  will  not  only  require  1.5  X  3  =  4.5  more,  but 


76  RAILROAD    CONSTRUCTION.  §  70. 

enough  additional  distance  so  that  the  added  distance  shall  be 
1.5  times  the  additional  drop.  As  a  first  trial  the  rod  may  be 
held  at  24  feet  out  and  a  reading  of,  say,  8.3  is  obtained.  8.3 
-|-  1.3  —  9.6,  the  depth  of  the  point  below  grade.  The  point 
on  the  slope  line  (n)  which  has  this  depth  below  grade  is  at  a 
-distance  from  the  center  a?  =8  +  1. 5x9. 6  =  22. 4.  The 
point  on  the  surface  (s)  having  that  depth  is  24  feet  out.  There- 
fore the  true  point  (m)  is  nearer  the  center.  A  second  trial  at 
20.5  feet  out  gives  a  rod  reading  of,  say,  7.1  or  a  depth.of  8.4 
below  grade.  This  corresponds  to  a  distance  out  of  20.6.  Since 
the  natural  soil  (especially  in  farming  lands  or  woods)  is  generally 
so  rough  that  a  difference  of  elevation  of  a  tenth  or  so  may  be 
readily  found  by  slightly  varying  the  location  of  the  rod  (even 
though  the  distance  from  the  center  is  the  same),  it  is  useless  to 
attempt  too  much  refine'ment,  and  so  in  a  case  like  the  above  the 
combination  of  8.4  below  grade  and  20.6  out  from  center  may 
be  taken  to  indicate  the  proper  position  of  the  slope-stake.  This 
is  usually  indicated  in  the  form  of  a  fraction,  the  distance  out 
being  the  denominator  and  the  height  above  (or  below)  grade 
being  the  numerator ;  the  fact  of  cut  or  fill  may  be  indicated  by 
C  or  F.  Ordinarily  a  second  trial  will  be  sufficient  to  determine 
with  sufficient  accuracy  the  true  position  of  the  slope-stake. 
Experienced  men  will  frequently  estimate  the  required  distance 
out  to  within  a  few  tenths  at  the  first  trial.  The  left-hand  pages 
of  the  note-book  should  have  the  station  number,  surface  eleva- 
tion, grade  elevation,  center  cut  or  fill,  and  rate  of  grade.  The 
right-hand  pages  should  be  divided  in  the  center  and  show  the 
distances  out  and  heights  above  grade  of  all  points,  as  is  illustrated 
in  §  84.  The  notes  should  read  UP  the  page,  so  that  when  look- 
ing ahead  along  the  line  the  figures  are  in  their  proper  relative 
position.  The  "fractions"  farthest  from  the  center  line  repre- 
sent the  slope-stake  points. 

COMPUTATION    OF    VOLUME. 

70.  Prismoidal  formula,     Let  Fig.  46  represent  a  triangular 
prismoid.     The  two  triangles  forming  the  ends  lie  in  parallel 


0. 


Y  1 
\G  X 


EARTHWORK. 


77 


planes,  but  since  the  angles  of  one  triangle  are  not  equal  to  the 
corresponding  angles  of  the  other  triangle,  at  least  two  of  the  sur- 
faces must  be  warped.  If  a  section,  parallel  to  the  bases,  is 


FIG.  46. 


made  at  any  point  at  a  distance  x  from  one  end,  the  area  of  the 
section  will  evidently  be 


Ax  = 


The  volume  of  a  section  of  infinitesimal  length  will  be  A^dx,  and 
the  total  volume  of  the  prismoid  will  be  * 


=  i&A*  +  (J,  -  J,)A,     +  J,(*.  -  A,) 


*  Students  unfamiliar  with  the  Integral  Calculus  may  take  for  granted  the 
fundamental  formulae  that   /  dx  =  x,  that   /  xdx  =  %x*,  and  that  /  x*dx  =  Jz8; 

also  that  in  integrating  between  the  limits  of  I  and  0  (zero),  the  value  of  the 
integral  may  be  found  by  simply  substituting  I  for  x  after  integration. 


78  RAILROAD  CONSTRUCTION.  §  70. 


,ft  +  A,) 


=  -  [^  +  4^  +  4],  ........     (45) 


in  which  -4, ,  -4. ,  and  A*  are  the  areas  respectively  of  the  two 
bases  and  of  the  middle  section.  Note  that  Am  is  not  the  mean 
of  ^and  A^ ,  although  it  does  not  necessarily  differ  very  greatly 
from  it. 

The  above  proof  is  absolutely  independent  of  the  values,  ab- 
solute or  relative,  of  b, ,  Z>2 ,  A,  or  A,.  For  example,  A2  may  be 
zero  and  the  second  base  reduces  to  a  line  and  the  prismoid  be- 
comes wedge-shaped ;  or  JA  and  A,  may  both  vanish,  the  second 
base  becoming  a  point  and  the  prismoid  reduces  to  a  pyramid 
Since  every  prismoid  (as  denned  in  §  67)  may  be  reduced  to  a 
combination  of  triangular  prismoids,  wedges,  and  pyramids,  and 
since  the  formula  is  true  for  any  one  of  them  individually,  it  is 
true  for  all  collectively ;  therefore  it  may  be  stated  that  * 

The  volume  of  a  prismoid  equals  one  sixth  of  the  perpendic- 
ular distance  between  the  bases  multiplied  ~by  the  sum  of  the 
areas  of  the  two  bases  plus  four  times  the  area  of  the  middle 
section. 

While  it  is  always  possible  to  compute  the  volume  of  any 
prismoid  by  the  above  method,  it  becomes  an  extremely  compli- 
cated and  tedious  operation  to  compute  the  true  value  of  the 
middle  section  if  the  end  sections  are  complicated  in  form.  It 

*  The  student  should  note  that  the  derivation  of  equation  (45)  does  not  com- 
plete the  proof,  but  that  the  statements  in  the  following  paragraph  are  logi- 
cally necessary  for  a  general  proof. 


§  12.  EARTHWORK.  79 

therefore  becomes  a  simpler  operation  to  compute  volumes  by  ap- 
proximate formulae  and  apply,  if  necessary,  a  correction.  The 
most  common  methods  are  as  follows : 

71.  Averaging   end   areas.     The   volume   of  the    triangular 
prismoid    (Fig.    46),    computed    by    averaging    end    areas,    is 

5"[i^A"HtWJ«  Subtracting  this  from  the  true  volume  (as 
2 

given  in  the  equation  above,  Eq.  (45)  ),  we  obtain  the  correction 


This  shows  that  if  either  the  A's  or  J's  are  equal,  the  correc- 
tion vanishes  ;  it  also  shows  that  if  the  bases  are  roughly  similar 
and  b  varies  roughly  with  A  (which  usually  occurs,  as  will  be 
seen  later),  the  correction  will  be  negative  ,  which  means  that  the 
method  of  averaging  end  areas  usually  gives  too  large  results. 

72.  Middle  areas.  Sometimes  the  middle  area  is  computed 
and  the  volume  is  assumed  to  be  equal  to  the  length  times  the 

middle  area.    This  will  equal  -  X    '  ~\     2  X  -     „     \     Subtract- 

a  a  1 

ing  this  from  the  true  volume,  we  obtain  the  correction 


(47) 


As  before,  the  form  of  the  correction  shows  that  if  either 
the  A's  or  J's  are  equal,  the  correction  vanishes;  also  under  the 
usual  conditions,  as  before,  the  correction  is  positive  and  only 
one-half  as  large  as  by  averaging  end  areas.  Ordinarily  the 
labor  involved  in  the  above  method  is  no  less  than  that  of 
applying  the  exact  prismoidal  formula. 

73.  Two-level  ground.  When  approximate  computations  of 
earthwork  are  sufficiently  exact  the  field-work  may  be  materi- 
ally reduced  by  observing  simply  the  center  cut  (or  fill)  and  the 


RAILROAD  CONSTRUCTION. 


§73. 


natural  slope  <*,  measured  with  a  clinometer.     The  area  of  such 
a  section  (see  Fig.  48)  equals 


FIG.  47. 


FIG.  48. 


But 

c 

from  which 

Similarly, 

Substituting, 


tan  ft  =  a  -\-  d  -f-  a?4  tan  or, 

_         a  +  <^ 
1  ~~  tan  /?  —  tan  a' 

a  -\-  d 
Xr  =  tan  ft  +  tan  a* 

tan  (3  ab 


Area  =  (a  +  d)*—-, 


tan2  /?  —  tan2  a        2  ' 


.     (48) 


The  values  a,  tan  /?,  tan9  /?  are  constant  for  all  sections,  so 
that  it  requires  but  little  work  to  find  the  area  of  any  section. 
As  this  method  of  cross-sectioning  implies  considerable  approxi- 
mation, it  is  generally  a  useless  refinement  to  attempt  to  com- 
pute the  volume  with  any  greater  accuracy  than  that  obtained 
by  averaging  end  areas.  It  may  be  noted  that  it  may  be  easily 
proved  that  the  correction  to  be  applied  is  of  the  same  form  as 
that  found  in  §  71  and  equals 


[(«%'+  O  -  K'  +  <')][(<*"+  «)  - 


<*)], 


§  74.  EARTHWORK.  81 

wliicli  reduces  to 


Correction^)  [(^M.£^-(^U.^.a.J[^-*]  [•  (49) 


"When  d"  =  d'  the  correction  vanishes.  This  shows  that 
when  the  center  heights  are  equal  there  is  no  correction  — 
regardless  of  the  slope.  If  the  slope  is  uniform  throughout, 
the  form  of  the  correction  is  simplified  and  is  invariably  nega- 
tive. Under  the  usual  conditions  the  correction  is  negative, 
i.e.,  the  method  generally  gives  too  large  results. 

74.  Level  sections.  When  the  country  is  very  level  or  when 
only  approximate  preliminary  results  are  required,  it  is  some- 
times assumed  that  the  cross-sections  are  level.  The  method  of 
level  sections  is  capable  of  easy  and  rapid  computation.  The 
area  may  be  written  as 

(a  +  d)*8  -~.       ......     (50) 


This  also  follows  from  Eq.  (48)  when  a  =  0  and  tan  fi  =  -. 

s 

s  here  represents  the  "slope  ratio,"  i.e.,  the  ratio  of  the  hori- 
zontal projection  of  the  slope  to  the  vertical.  A  table  is  very 
readily  formed  giving  the  area  in  square  feet  of  a  section  of 
given  depth  and  for  any  given  widtli  of  roadbed  and  ratio  of 
side-slopes.  The  area  may  also  be  readily  determined  (as  illus- 
trated in  the  following  example)  without  the  use  of  such  a 
table;  a  table  of  squares  will  facilitate  the  work.  Assuming 


82  RAILROAD  CONSTRUCTION.  §  75. 

the  cross-sections  at  equal  distances  (=  I)  apart,  the  total  ap- 
proximate volume  for  any  distance  will  be 

2 

The  prismoidal  correction  may  be  directly  derived  from 
Eq.  (46)  as  ^[2(0  +  d')s  -  2(a  +  d")s][(a  +  d")  -  (a  +  d']\y, 
which  reduces  to 

-l^(d'-d'J     or      -^-(d'-d")\        .     (52) 

This  may  also  be  derived  from  Eq.  (49),  since  a  =  0, 
tan  OL  —  0,  and  tan  ft  =  %a  -=-  &.  This  correction  is  always 
negative,  showing  that  the  method  of  averaging  end  areas, 
when  the  sections  are  level,  always  gives  too  large  results.  The 
prismoidal  correction  for  any  one  prismoid  is  therefore  a  con- 
stant times  the  square  of  a  difference.  The  squares  are  always 
positive  whether  the  differences  are  positive  or  negative.  The 
correction  therefore  becomes 


75.  Numerical  example :  level  sections.  Given  the  following 
center  heights  for  the  same  number  of  consecutive  stations  100 
feet  apart;  width  of  roadbed  18  feet;  slope  1J  to  1. 

The  products  in  the  fifth  column  may  be  obtained  very 
readily  and  with  sufficient  accuracy  by  the  use  of  the  slide-rule 
described  in  §  79.  The  products  should  be  considered  as 

(a+  d)(a  +  d)  -r-  -.       In    this   problem    s  =  !£,-  =  .6667. 
s  s 

To  apply  the  rule  to  the  first  case  above,  place  6667  on  scale  I> 
over  89  on  scale  A,  then  opposite  89  on  scale  B  will  be  found 


§76 


EARTHWORK. 


83 


118.8  on  scale  A.  The  position  of  the  decimal  point  will 
be  evident  from  an  approximate  mental  solution  of  the  prob- 
lem. 


1 
Sta. 

Center 
Height. 

a  +  d 

(a  +  d)» 

(a  +  d)*s 

Areas. 

d'  ~  d" 

(d'  ~  d")2 

17 
18 
19 
20 
21 
22 

2.9 
4.7 
6.8 
11.7 
4.2 
1.6 

8.9 
10.7 
12.8 
17.7 
10.2 
7.6 

79.21 
114.49 
163.84 
313.29 
104.04 
57.76 

118.81 

171.741 
245.761 
469.93  f 
156.  06  J 
86.64 

118.81 
T343.48 
2_  1491.52 
x<5-]  939.86 
L312.12 
86.  -64 

1.8 
2.1 
4.9 
7.5 
2.6 

3.24 
4.41 
24.01 
56.25 
6.76 

<ib      6  X  18 


=  54 

1752.43  X  100 

2X27 

Corr'  =  - 


2292.43 
10X54=    540 

1752.43 

=  3245  cub.  yards     =  approx.  vol. 


94.67 


OT  x  9467      =  -  91  cub'  yds' 

3245  —  91  =  3154  cub.  yds.         =  exact  volume. 

The  above  demonstration  of  the  correction  to  be  applied  to 
the  approximate  volume,  found  by  averaging  end  areas,  is  intro- 
duced mainly  to  give  an  idea  of  the  amount  of  that  correction. 
Absolutely  level  sections  are  practically  unknown,  and  the  error 
involved  in  assuming  any  given  sections  as  truly  level  will 
ordinarily  be  greater  than  the  computed  correction.  If  greater 
accuracy  is  required,  more  points  should  be  obtained  in  the 
cross-sectioning,  which  will  generally  show  that  the  sections 
are  not  truly  level. 

76.  Equivalent  sections.  When  sections  are  very  irregular 
the  following  method  may  be  used,  especially  if  great  accuracy 
is  not  required.  The  sections  are  plotted  to  scale  and  then  a 
uniform  slope  line  is  obtained  by  stretching  a  thread  so  that  the 
undulations  are  averaged  and  an  equivalent  section  is  obtained. 
The  center  depth  (d)  and  the  slope  angle  (a)  of  this  line  can 
be  obtained  from  the  drawing,  but  it  is  more  convenient  to 
measure  the  distances  (a?/  and  #r)  from  the  center.  The  area 


84  RAILROAD   CONSTRUCTION.  §  76 


may  then  be   obtained   independent   of   the   center    depth   as 
follows  :  Let  s  =  the  elope 
(See  Fig.  48.)     Then  the 


follows  :  Let  s  =  the  elope  ratio  of  the  side  slopes  =  cot  ft  =  ^-  . 


db 

j (54> 


The  true  volume,  according  to  the  prismoidal  formula,  of  a 
length  of  the  road  measured  in  this  way  will  be 

I  pp/ay.'  _  ab         (xj  +  x{'  xr'  +  xr"  l__ab\       xj'x^'  _  ab-, 
6|_    s       ~  2  +    \       2  2         s    ~  2/"1       s        "  2  j* 

If  computed  by  averaging  end  areas,  the  approximate  volume 
will  be 

Z  faa'      d$       a'a" 


.       «   "~* 


Subtracting  this  result  from  the  true  volume,  we  obtain  as  the 
correction 

Correction  =  T-,(#/'  —  #/)(#/  —  av").     •      •      (55) 


This  shows  that  if  the  side  distances  to  either  the  right  or 
left  are  equal  at  adjacent  stations  the  correction  is  zero,  and 
also  that  if  the  difference  is  small  the  correction  is  also  small 
and  very  probably  within  the  limit  of  accuracy  obtainable  by 
that  method  of  cross-sectioning.  In  fact,  as  has  already  been 
shown  in  the  latter  part  of  §  75,  it  will  usually  be  a  useless 
refinement  to  compute  the  prismoidal  correction  when  the 
method  of  cross-sectioning  is  as  rough  and  approximate  as  this- 
method  generally  is. 


§  77.  EARTHWORK.  85 

77.  Equivalent  level  sections.  These  sloping  "  two-level" 
sections  are  sometimes  transformed  into  ' '  level  sections  of  equal 
area, ' '  aad  the  volume  computed  by  the  method  of  level  sections 
(§  74).  But  the  true  volume  of  a  prismoid  with  sloping  ends  does 
not  agree  with  that  of  a  prismoid  with  equivalent  bases  and  level 
ends  except  under  special  conditions,  and  when  this  method  is 
used  a  correction  must  be  applied  if  accuracy  is  desired,  although, 
as  intimated  before,  the  assumption  that  the  sections  have  uni- 
form slopes  will  frequently  introduce  greater  inaccuracies  than 
that  of  this  method  of  computation.  The  following  demonstra- 
tion is  therefore  given  to  show  the  scope  and  limitations  of  the 
errors  involved  in  this  much  used  method. 

In   Fig.   50,    let  dl  be  the  center  height  which  gives   an 


FIG.  50. 
equivalent  level  section.     The  area  will  equal  (a  -f-  d^s  —  -^-, 

/V3  rp  fjf)  f) 

which  must  equal  the  area  given  in  §76,  — ~-.     s  =  x-. 


( '  ~    \    j  \*  ff 

(*+«•)«=   --, 


or       a 


To  obtain  dl  directly  from  notes,  given  in  terms  of  d  and  #, 


36  RAILROAD  CONSTRUCTION.  §77. 

we  may  substitute  the  values  of  xt  and  xr  given  in  §  73,  which 
gives 

tan  ft  _          a+  d 

=  '       *' 


-  tan 


The  true  volume  of  the  equivalent  section  may  be  repre- 
sented by 


From  this  there  should  be  subtracted  the  volume  of  the 
<  '  grade  prism  '  '  under  the  roadbed  to  obtain  the  volume  of  the 
cut  that  would  be  actually  excavated,  but  in  the  following  com- 
parison, as  well  as  in  other  similar  comparisons  elsewhere  made, 
the  volume  of  the  grade  prism  invariably  cancels  out,  and  so  for 
the  sake  of  simplicity  it  will  be  disregarded.  This  expression 
for  volume  may  be  transposed  to 


,  , 

J  '          "        - 


6L    s9  \     2s  2s 

The  true  volume  of  the  prismoid  with  sloping  ends  is  (see 
76) 

l_ 
6 


r^v  ,  4n*L±*R(<±*fil\  L *  "^"1 
L  s     \\    2    A    2    y?/  -  ~T" 


The  difference  of  the  two  volumes 


=  ~<  V^Wf-  VxS'XrJ. (58) 

This    shows  that    "  equivalent    level    sections"   do  not  in 
general  give  the  true  volume,  there  being  an  exception  when 


§  78.  EARTHWORK.  87 

,/-, './•/'  =  #/'#/.  This  condition  is  fulfilled  when  the  slope  is 
uniform,  i.e.,  when  a.'  =  a".  When  this  is  nearly  so  the  error 
is  evidently  not  large.  On  the  other  hand,  if  the  slopes  are  in- 
clined in  opposite  directions  the  error  may  be  very  considerable, 
particularly  if  the  angles  of  slope  are  also  large. 

78.  Three-level  sections.     The  next  method  of  cross-section- 
ing in  the  order  of  complexity,  and  therefore  in  the  order  of 


FIG.  51. 


accuracy,  is  the  method  of  three-level  sections.     The  area  of  the 
section    is   %(a  +  d)(wr  +  wt)  --  ,    which    may   be    written 

Zt 

\(a  +  d)w  --   »   i°   wnich  w  =  wr  +  w^     If  the  volume   is 


computed  by  averaging  end  areas,  it  will  equal 
Z 


_ 


f  -  db  +  (a  +  d"}w"  —  db"\.    .     .     (59) 


If  we  divide  by  27  to  reduce  to  cubic  yards,  we  have,  when 
I  =  100, 


For  the  next  section 


88  RAILROAD   CONSTRUCTION.  §  78. 

For  a  partial  station  length  compute  as  usual  and  multiply 

length  in  feet       ml  .        .  ,  , 

result  by  --  .     Ihe    pnsmoidal  correction  may  be 

obtained  by  applying  Eq.  (46)  to  each  side  in  turn.     For  the  left 
side  we  have 

-[(0  +  df)  -  (a  +  d")]  W  ~  wfii     which  e(luals 


For  the  right  side  we  have,  similarly, 

L(d'  -  d")(wr"  -  <). 

The  total  correction  therefore  equals 


Reduced  to  cubic  yards,  and  with  I  =  100, 

- 

Pris.  Corr.  =  ff(^  -  d")(w"—u/). 


When  this  result  is  compared  with  that  given  in  Eq.  (55) 
there  is  an  apparent  inconsistency.  If  two-level  ground  is  con- 
sidered as  but  a  special  case  of  three-level  ground,  it  would  seem 
as  if  the  same  laws  should  apply.  If,  in  Eq.  (55),  xr'  =  a?r", 
and  a?/'  is  different  from  a?/,  the  equation  reduces  to  zero  ;  but 
in  this  case  d'  would  also  be  different  from  d"  ;  and  since  x{  + 
xr'  would  =  w',  and  x{'  +  xr"  =  w"  m  Eq.  (60),  w"  —  w'  would 
not  equal  zero  and  the  correction  would  be  some  finite  quantity 
and  not  zero.  The  explanation  lies  in  the  difference  in  the  form 
and  volume  of  the  prismoids,  according  to  the  method  of  the 


78. 


EARTHWORK. 


formation  of  the  warped  surfaces.  If  the  surface  is  supposed  to 
be  generated  by  the  locus  of  a  line  moving  parallel  to  the  ends 
as  plane  directors  and  along  two  straight  lines  lying  in  the  side- 
slopes,  then  #jmid-  will  equal  i(a?/  +  a?/7),  and  a?rmid'  will  equal 
%(xr'  -f-  xr"),  but  the  profile  of  the  center  line  will  not  be 
straight  and  dmld-  will  not  equal  J(rZ'  +  d").  On  the  other 
hand,  if  the  surfaces  be  generated  by  two  lines  moving  parallel 
to  the  ends  as  plane  directors  and  along  a  straight  center  line 
and  straight  side  lines  lying,in  the  slopes,  a  warped  surface  will 
be  generated  each  side  of  the  center  line,  which  will  have  uni- 
form slopes  on  each  side  of  the  center  at  the  two  ends  and  no- 
where  else.  This  shows  that  when  the  upper  surface  of  earth- 
work is  warped  (as  it  generally  is),  two-level  ground  should  not 
be  considered  as  a  special  case  of  three-level  ground.  This  dis- 
cussion, however,  is  only  valuable  to  explain  an  apparent  incon- 
sistency and  error.  The  method  of  two-level  ground  should 
only  be  used  when  such  refinements  as  are  here  discussed  are  of 
no  importance  as  affecting  the  accuracy. 

The  following  example  is  given  to  illustrate  the  method  of 
three-level  sections. 


H 
1 

17 
18 
+40 
19 
20 

Center. 

I 

i 
a 

a  +  d 

IV 

Yards. 

d>  -  d" 

MO"-W' 

11 

-20 
-  3 
-11 
1S 

*I~*r 

V(xrxr) 

11 

+4 

-H 

+5 
+3 

3R 

+1 
+3 
+6 
+2 
+1 

2.6F 
SAF 
10.7F 
GAF 
3.7F 

10.6F 

0.8F 

7.3 
12.8 
15.4 
11.1 
8.4, 

31.1 
42.8 
51.5 
38.1 
23.0 

210 
507 
734 
392 
179 

595 
448 
602 
449 

-5.5 
-2.6 

+4.3 
+2.7 

+11.7 
+  8.7 
13  4 

14.7 
18.6 
23.1 
17.9 
8.4 

a-,'.  9 

15.SF 

8.2 
3.4F 

30.7 
00.2F 

12.  1 
4.8F 

37.3 
14.0F 

14.2 
2.1F 

¥8.0 
S.8F 

10.1 
0.2F 

1C     \ 

15.7 

7.3 

Roadbed,  14'  wide  in  fill.    Approx.  Vol.=2094 
Slope  1^  to  1.  Pris.  corr.      =    47 


-47 


True  Vol.        =2047  (disregarding  curv.  corr).* 

*  For  the  Derivation  of  the  curvation  correction,  see  §  93. 


90  RAILROAD   CONSTRUCTION.  §79. 

In  the  first  column  of  yards 

210  =  ffO  +  d}w  =  f4  X  7.3  x  31.1 ; 
507,  734,     etc.,  are  found  similarly; 
595  =  210  -  61  +  507  --  61 ; 

448  =  T4o°<r(5°7  —  61  +  734  —  61); 
602  =  T6o°o(734  -  61  +  392  -  61); 

449  =  392-61  +  179-  61. 
For  the  prismoidal  correction, 

_  20  =  l\(df  -  d")(w"  -  w')  =  jf(2.6  -  8.1)(42.8  -  31.1) 


For  the  next  line,  -  3  =  yVoBK-  2.8)(+ 8.7)],  and 
similarly  for  the  rest.  The  "  F"  in  the  columns  of  center 
heights,  as  well  as  in  the  columns  of  "  right  "  and  "  left,"  are 
inserted  to  indicate  fill  for  all  those  pooints.  Cut  would  be 
indicated  by  "  <7." 

79.   Computation    of  products,     The   quantities   -^(a  +  d)w 

and  db   represent  in  each  case  the   product  of  two  variable 

2  7 

terms  and  a  constant.  These  products  are  sometimes  obtained 
from  tables  which  are  calculated  for  all  ordinary  ranges  of  the 
variable  terms  as  arguments.  A  similar  table  computed  for 

_((lr  —  d")(w"  —  w')  will  assist  similarly  in  computing  the 
81 

prismoidal  correction.  Prof.  Charles  L.  Crandall,  of  Cornell 
University,  is  believed  to  be  the  first  to  prepare  such  a  set  of 
tables,  which  were  first  published  in  1886  in  "Tables  for  the 
Computation  of  Railway  and  Other  Earthwork."  Another 


§  79.  EARTHWORK.  91 

easy  method  of  obtaining  these  products  is  by  the  use  of  a  slide- 
rule.  A  slide-rule  has  been  designed  by  the  author  to  accom- 
pany this  volume.  It  is  designed  particularly  for  this  special 
work,  although  it  may  be  utilized  for  many  other  purposes  for 
which  slide-rules  are  valuable.  To  illustrate  its  use,  suppose 
(a  +  d)  =  28.2,  andw  =  62.4;  then 


25,     .     ,.          28.2  X  62.4 


27V  1.08 

Set  108  (which,  being  a  constant  of  frequent  use,  is  specially 
marked)  on  the  sliding  scale  (I>)  opposite  282  on  the  other  scale 
(A),  and  then  opposite  624  on  scale  B  will  be  found  1629  on 
scale  A,  the  162  being  read  directly  and  the  9  read  by  estima- 
tion. Although  strict  rules  may  be  followed  for  pointing  off 
the  final  result,  it  only  requires  a  very  simple  mental  calculation 
to  know  that  the  result  must  be  1629  rather  than  162.9  or 
16290.  For  products  less  than  1000  cubic  yards  the  result 
may  be  read  directly  from  the  scale;  for  products  between  1000 
and  5000  the  result  may  be  read  directly  to  the  nearest  10 
yards,  and  the  tenths  of  a  division  estimated.  Between  5000  and 
10,000  yards  the  result  may  be  read  directly  to  the  nearest  20 
yards,  and  the  fraction  estimated ;  but  prisms  of  such  volume 
will  never  be  found  as  simple  triangular  prisms — at  least,  an  as- 
sumption that  any  mass  of  ground  was  as  regular  as  this  would 
probably  involve  more  error  than  would  occur  from  faulty  esti- 
mation of  fractional  parts.  Facilities  for  reading  as  high  as 
10,000  cubic  yards  would  not  have  been  put  on  the  scale  ex- 
cept for  the  necessity  of  finding  such  products  as  f  |(9.1  X  9.5), 
for  example.  This  product  would  be  read  off  from  the  same 
part  of  the  rule  as  ff (91  X  95).  In  the  first  case  the  product 
(80.0)  could  be  rea$  directly  to  the  nearest  .2  of  a  cubic  yard, 
which  is  unnecessarily  accurate.  In  the  other  case,  the  prod- 
uct (8004)  could  only  be  obtained  by  estimating  -^  of  a  division. 
The  computation  for  the  prismoidal  correction  may  be  made 


92  RAILROAD  CONSTRUCTION.  §  80. 

similarly  except  that  the  divisor  is  3.24  instead  of  1.08.      For 
example,  |f(5.5  X  11.7)  =  ^^  ^J .     Set  the  324  on  scale 

O..24 

B  (also  specially  marked  like  108)  opposite  55  on  scale  A,  and 
proceed  as  before. 

80.  Five-level  sections.     Sometimes  the  elevations  over  each 
edge  of  the  roadbed  are  observed  when  cross-sectioning.      These 
are  distinctively  termed   u  five  -level  sections."     If  the  center, 
the  slope-stakes,  and  one  intermediate  point  on  each  side  (not 
necessarily  over   the  edge  of  the   roadbed)  are  observed,   it  i-s 
termed  an  "  irregular  section."     The  field-work  of  cross-section- 
ing five-level  sections  is  no  less  than  for  irregular  sections  with 
one  intermediate  point;  the  computations,  although  capable  of 
peculiar  treatment  on  account  of  the  location  of  the  intermediate 
point,  are  no  easier,  and  in  some  respects  more  laborious;  the 
cross-sections  obtained  will  not  in  general  represent  the  actual 
cross- sections  as  truly  as  when  there  is  perfect  freedom  in  locat- 
ing the  intermediate  point;   as  it  is  generally  inadvisable  or  un- 
necessary to  employ  five-level  sections  throughout  the  length  of 
a  road,  the  change  from  one  method  to  another  adds  a  possible 
element  of  inaccuracy  and  loses  the  advantage  of  uniformity  of 
method,   particularly  in  the  note's  and  form  of  computations. 
On  these  accounts  the  method  will  not  be  further  developed, 
except  to  note  that  this  case,   as  well  as   any  other,  may  be 
solved  by  dividing  the  whole  prismoid  into  triangular  prismoids, 
computing  the  volume  by  averaging  end  areas,  and  computing 
the  prismoidal  correction  by  adding  the  computed  corrections 
for  each  elementary  triangular  prismoid. 

81.  Irregular    sections.     In    cross-sectioning    irregular    sec- 
tions, the  distance  from  the   center  and   the  elevation  above 
"grade"    of   every    "break"    in    the    cross-section    must    be 
observed.     The  area  of  the  irregular  section  may  be  obtained 
by  computing  the  area  of  the  trapezoids  (five,  in  Fig.  44)  and 
subtracting  the  two  external  triangles.     For  Fig.  44  the  area 
would  be 


§81. 


EARTHWORK. 

d+jr          jr 


93 


r 

(2/r  -  2r 


FIG.  44. 


Expanding  this  and  collecting  terms,  of  which  many  will 
cancel,  we  obtain 


AEEA  =  - 


r(jr  - 


.   .     .     (61) 


An  examination  of  this  formula  will  show  a  perfect  regu- 
larity in  its  formation  which  will  enable  one  to  write  out  a 
similar  formula  for  any  section,  no  matter  how  irregular  or  how 
•many  points  there  are,  without  any  of  the  preliminary  work. 
The  formula  may  be  expressed  in  words  as  follows  : 

AREA  equals  one-half  the  sum  of  products  obtained  as  follows  : 

the  distance  to  each  slope-  stake  times  the  height  above  grade 
of  the  point  next  inside  the  slope-stake  ; 

tJie  distance  to  each,  intermediate  point  in  turn  times  the  height 
of  the  point  just  inside  minus  the  height  of  the  point  just  outside  j 

finally,  one-half  the  width  of  the  roadbed  times  the  sum  of 
the  slope-  stake  heights. 


94 


RAILROAD  CONSTRUCTION. 


§82. 


If  one  of  the  sides  is  perfectly  regular  from  center  to  slope- 
stake,  it  is  easy  to  show  that  the  rule  holds  literally  good. 
The  "  point  next  inside  the  slope- stake "  in  this  case  is  the 
center;  the  intermediate  terms  for  that  side  vanish.  The  last 
term  must  always  be  used.  The  rule  holds  good  for  three -level 
sections,  in  which  case  there  are  three  terms,  which  may  be 
reduced  to  two.  Since  these  two  terms  are  both  variable  quan- 
tities for  each  cross-section,  the  special  method,  given  in  §  78, 

in  which  one  term  ( —  J  is  a  constant  for  all  sections,  is  pref- 
erable. In  the  general  method,  each  intermediate  u  break  " 
adds  another  term. 

82,  Volume  of  an  irregular  prismoid.  If  there  is  a  break  at 
one  cross-section  which  is  not  represented  at  the  next,  the  ridge 
(or  hollow)  implied  by  that  break  is  supposed  to  ' '  vanish ' '  at 
the  next  section.  In  fact,  the  volume  will  not  be  correctly 


FIG.  52. 

represented  unless  a  cross-section  is  taken  at  the  point  where 
the  ridge  or  hollow  "  vanishes"  or  "  runs  out."  To  obtain 
the  true  prismoidal  correction  it  is  necessary  to  observe  on  the 
ground  the  place  where  a  break  in  an  adjacent  section,  which 
is  not  represented  in  the  section  being  taken,  runs  out.  For 
example,  in  Fig.  52,  the  break  on  .the  left  of  section  A'1 ',  at  a 


§  83.  EARTHWORK.  95 

distance  of  y/'from  the  center,  is  observed  to  run  out  in  section 
A'  at  a  distance  of  y{  from  the  center.  The  volume  of  the 
prismoid,  computed  by  the  prismoidal  formula  as  in  §  TO,  will 
involve  the  midsection,  to  obtain  the  dimension  of  which  will 
require  a  laborious  computation.  A  simpler  process  is  to  compute 
the  volume  by  averaging  end  areas  as  in  §  81  and  apply  a 
prismoidal  correction.  To  do  this  write  out  an  expression  for 
each  end  area  similar  to  that  given  in  Eq.  61.  The  sum  of 

these  areas  times  ~-  gives  the  approximate  volume.     As  before, 

2t 

length  in  feet 
for  partial  station  lengths,  multiply  the  result  by  - 


There  will  be  no  constant  subtractive  term,  f  f#£,  as  in  §  78. 
The  true  prismoidal  correction  may  be  computed,  as  in  §  83,  or 
the  following  approximate  method  may  be  used  :  Consider  the 
irregular  section  to  be  three-level  ground  for  the  purpose  of 
computing  the  connection  only.  This  has  the  advantage  of  less 
labor  in  computation  than  the  use  of  the  true  prismoidal  correc- 
tion, and  although  the  error  involved  may  be  considerable  in 
individual  sections,  the  error  is  as  likely  to  be  positive  as  nega- 
tive, and  in  the  long  run  the  error  will  not  be  large  and  generally 
will  be  much  less  than  would  result  by  the  neglect  of  any 
prismoidal  correction. 

83.  True  prismoidal  correction  for  irregular  prismoids.  As 
intimated  in  §  82,  each  cross-  section  should  be  assumed  to  have 
the  same  number  of  sides  as  the  adjacent  cross-  section  when 
computing  the  prismoidal  correction.  This  being  done,  it  per- 
mits the  division  of  the  whole  prismoid  into  elementary  triangu- 
lar prismoids,  the  dimensions  of  the  bases  of  which  being  given 
in  each  case  by  a  vertical  distance  above  grade  line  and  by  the 
horizontal  distance  between  two  adjacent  breaks.  The  summa- 
tion of  the  prismoidal  corrections  for  each  of  the  elementary 
triangular  prismoids  will  give  the  true  prismoidal  correction. 
Assuming  for  an  example  the  cross-section  of  Fig.  44,  with  a 
cross-section  of  the  same  number  of  sides,  and  with  dimensions 


96  RAILROAD   CONSTRUCTION.  §83. 

similarly  indicated,  for  the  other  end,  the  prismoidal  correction 
becomes  (see  Eq.  46) 


-  yi)  +  (d'  -  d"W  -  yi')  +  (d'  -  d")(zr"  -  zr'} 

jr")(Zr"  -  *r)  +  (jr    ~  jr")[(yr"  -  Zr")  -   (yr'  -  «rO] 
kr")[(yr"  -  Zr")  -  (yr    -  2/)l 


-  «  -      -cv-  v 


Expanding  this  and  collecting  terms,   of  which  many  will 
cancel,  we  obtain 


Pris.  Corr.  =         aS"  -aji')(*i'  -  *i")  +  W 

*r'    -  *r")  +  (yr"  -  y/)[(>'  -  V)  -  (jr"  ~  V)] 

-V)-(d;/-  V)]]  ........     (62) 

By  comparing  this  equation  with  Eq.  61  a  remarkable 
coincidence  in  the  law  of  formation  may  be  seen,  which  enables 
this  formula  to  be  written  by  mere  inspection  and  to  be  applied 
numerically  with  a  miniitfum  of  labor  from  the  computations  for 
end  areas,  as  will  be  shown  (§  84)  by  a  numerical  example. 
For  each  term  in  Eq.  61,  as,  for  example,  yr(jr  —  ^r)>  there  is 
a  correction  term  in  Eq.  62  of  the  form 


/   -  V)  -    (jr"   ~    V')]. 


Each  one  of  these  terms  (yrx/,  y/,  (jr  —  A/),  and  (jr"  —  V)  ) 
has  been  previously  used  in  finding  the  end  areas  and  has  its 
place  in  the  computation  sheet.  The  summation  of  the  products 
of  these  differences  times  a  constant  gives  the  total  true  pris- 
moidal correction  in  cubic  yards  for  the  whole  prismoid  considered. 
The  constant  is  the  same  as  that  computed  in  §  78,  i.e.,  ||. 


§84. 


EARTHWORK. 


97 


84.  Numerical  example ;  irregular  sections ;  volume,  with  true 
prismoidal  correction. 


Sta. 

I  cut  - 
Centers  or 
{fill. 

Left. 

Right. 

19 

0.6e 

3.6c        /2.3c\         /1.8c\ 
1474        *872/         '  670/ 

O.lc          0.4c 
472            9TF 

18 

2.3c 

4.2c  »,„    6.8c           3.2e 
15.3            8.4             5.2 

/1.9c\         1.2c 
V3.6/        TO 

17 

7.6c 

8.2e         10.  2c          8.0c 

/5.8<-\        4.2c 

21.3           17.4            6.1 

\8.0/        15.3 

+  42 

10.  2c 

12.  2c       /12.3<-\         12.  6c 

6.2c          8.4c 

27.3         \22.07           8.2 

7.5           21.6 

16 

6.8c 

8.9<r                            7.6c 
2274                              12.0 

3.2c          2.6c 
4.1           12.9 

Koadbed  18  feet  wide  in  cut;  slope  1£  to  1. 

(12.3c\ 
'  Q)  mean  that  it  was  noted  in 

the  field  that  the  break,  indicated  at  Sta.  17  as  being  17.4  to 
the  left,  ran  out  at  Sta.  16  +  42  at  22.0  to  the  left.  By  inter- 
polation between  8.2  and  27.3  the  height  of  this  point  is 
computed  as  12.3.  The  quantities  in  the  other  brackets  are 
obtained  similarly.  These  quantities  are  only  used  when  the 
computation  of  the  true  prismoidal  correction  is  desired.  They 
are  not  needed  in  computing  the  volume  by  averaging  end 
areas,  nor  are  they  used  at  all  if  the  prismoidal  correction  is  to 
be  obtained  by  assuming  (for  this  purpose)  the  ground  to  be 
three-level  ground. 

In  the  tabular  form  on  page  98  the  figures  within  the  braces 
( — v- — •)  are  NOT  used  in  computing  the  volume,  but  are  only 
used  to  obtain  the  differences  of  widths  or  heights  with  which  to 
compute  the  true  prismoidal  correction.  It  may  be  noted,  as  a 
check,  that  the  volume,  computed  from  these  figures  in  the 
braces,  is  the  same  as  that  computed  from  the  other  figures. 


9§  RAILROAD   CONSTRUCTION.  §  84. 

VOLUME  OF  IRREGULAR  PRISMOID,  WITH  TRUE  PRISMOIDAL  CORRECTION. 


True  pris.  corr. 

Cf  0 

Width. 

Height. 

Yards. 

to"  —  w' 

h'  -  h" 

Yards. 

r-22.4 

7.6 

158 

LI  2.0 

-  2.1 

-23 

16 

12.9"JT> 

3.2 

40 

4.lJ 

4.2 

16 

9^0 

11.5 

96 

TT27.3 
LL  8.2 

12.6 
-  2.0 

~~319~ 
-15 

TTir 

-3.8 

-5.0 
-0.1 

-7 
0 

I  27.3 

12.3 

+  42 

L-^22.0 

(    8.2 

0.4 
-2.1 

21.  6]  R 
7.5JK 

6.2 
1.8 

124 
13 

+  8.7 
+  3.4 

-30 

+  2.4 

-8 

+  3 

9.0 

20.6 

172 

378 

(-5) 

r21.3 
L    17.4 

Le.i 

10.2 
-  0.2 
-  2.6 

201 
-    3 
-14 

-6.0 
-  4.6 
-2.1 

+  2.1 
+  0.6 
+  0.5 

-4 
-  1 
0 

17 

15.3)R 

5.8 

-  6.3 

+  0.4 

-I 

80) 

3.4 

-t-0.5 

-1.6 

0 

15'.3]R 

7  6 

107 

9.0 

12.4 

103 

584 

(-3) 

[-15.3 
L      8.4 
L  5.2 

6.8 
-  1.0 
-4.5 

95 

-    7 
-22 

-6.0 
-9.0 
-0.9 

+  3.4 

+  0.8 
+  1.9 

-  6 

-2 

18 

10.8]R 

2.3 

23 

-4.5 

+  5.3 

—  7 

10.8  In 

1.9 

3.6  f  "' 

1.1 

9.0 

5.4 

45 

528 

(-16) 

L[14.4 

0.6 

8 

(14.4 

2.3 

-0.9 

+  4.5 

-  1 

LI    8.2 

-  1.8 

-0.2 

+  0.8 

0 

19 

(    6.0 

-  1.7 

+  0.8 

-2.8 

-  1 

9.6]R 
4.2|R 

0.1 
0.2 

1 
1 

-1.2 
+  0.6 

+  1.8 
+  0.9 

-  1 
0 

9.0 

4.0 

33 

177 

(-3) 

Approx.  vol.       =1667                                      —  27 

True  pris.  corr.  =  —  27 

True  volume      =   1640  cubic  yards 

The  figures  within  each  brace  (or  bracket)  constitute  a  group 
which  must  be  used  in  connection  with  a  group  which  has  the 
same  number  of  points,  on  the  same  side  of  the  center,  in  the 
next  cross-section,  previous  or  succeeding.  In  the  column  of 


§  86.  EARTHWORK.  99 

4 'Yards"   under  u  True   pris.  corr.,"  we  have,   for  example, 


85.  Volume  of  irregular  prismoid,  with  approximate  prismoidal 
correction.  If  the  prismoidal  correction  is  obtained  approxi- 
mately, by  the  method  outlined  in  §  82,  the  process  will  be  as 
shown  in  the  tabular  form.  Not  only  is  the  numerical  work 
considerably  less  than  the  exact  method,  but  the  discrepancy  in 
cubic  yards  is  almost  insignificant. 


Sta. 

Width. 

Height. 

Yards. 

Cen. 
Height. 

Total 
width. 

d'-d" 

V)"—  W1 

Approx. 
pris.  corr. 

16 

+  42 

22.4 
12.0 
12  9 
4.1 
9.0 

IZTLZ 

8.2 
21.6 
7.5 
9.0 

7.6 
-2.1 
3.2 
4.2 
11.5 

158 
-  23 
40 
16 
96 

378 

+  6.8 

TToT2~ 

35.3 

-  14 

(-6) 

12.6 
-2.0 
6.2 
1.8 
20.6 

319 
-  15 
124 
13 
172 

48.9 

-3.4 

+  13.6 

17 

18 

21.3 
17.4 
6.1 
15  3 
9.0 

15.3 
8.4 
5.2 
10.8 
9.0 

10.2 
-0.2 
-2.6 
7.6 
12.4 

~6~1T 
-1.0 
-4.5 
2.3 
5.4 

201 
-    3 
-  14 
107 
103 

~95~ 
-    7 
-  22 
23 
45 

584 

+  7.6 
+  2.3 

36.6 

+2.6 
"+5^3" 

-12.3 

-10 

(-6) 
-17 

(-17) 

528 

26.1 

-  10.5 

19 

14.4 
9.6 
4.2 
9.0 

0.6 
0.1 
0.2 
4.0 

8 
1 
1 
33 

177 

+  0.6 

24.0 

+1.7 

-2.1 

-1 

(-D 

Approx.  volume      =    1667  —  30 

Approx.  pris.  corr.  =  —  30 

Corrected  volume    =   1637  cubic  yards 

86.  Illustration  of  value  of  approximate  rules.  The  accom- 
panying tabulation  shows  that  when  the  volume  of  an  irregular 
prismoid  is  computed  by  averaging  end  areas  and  is  corrected 
by  considering  the  ground  as  three-level  ground  (for  the  pur- 


100    .  RAILROAD  CONSTRUCTION.  §  87. 

poses  of  the  correction  only),  the  error  for  the  different  sections 
is  sometimes  positive  and  sometimes  negative,  and  in  this  case 


b£ 

£13 

•   05    > 

:     i*      a» 

$> 

Iff 

o   . 

B-S 

ill. 

Sections. 

'o 

X*  >  e3 

s& 

M°S  a 

Error. 

fc'  a.  °  "7  * 

Error. 

True  v 

OfO'  33 

B  ••  i» 

£11 

5 

B  |j«s9 

^.^^2 
a«o  bo 
^ 

8S  1-e.^ 
^S^  °  - 

16..           16  +  42 

373 

378 

-    5 

-     6 

—  1 

396 

+  23 

16  +  42...  17 

581 

584 

-     3 

-     6 

-  3 

577. 

-     4 

17.7.  18 

512 

528 

-  16 

-  17 

-   1 

463 

-  49 

18     19 

174 

177 

-     3 

-     1 

+  2 

147 

—  27 

1640 

1667 

-  27 

-  30 

-   3 

1583 

-  57 

amounts  to  only  3  yards  in  1640 — less  than  ^  of  1#.  If  the 
prismoidal  correction  had  been  neglected,  the  error  would  have 
been  27  yards — nearly  2$.  The  approximate  results  are  here 
too  large  for  each  section — as  is  usually  the  case.  If  points 
between  the  center  and  slope  stakes  are  omitted  and  the  volume 
computed  as  if  the  ground  were  three-level  ground,  the  error  is 
quite  large  in  individual  sections,  but  the  errors  are  both  posi- 
tive and  negative  and  therefore  compensating. 

87.  Cross-sectioning  irregular  sections,     The  prismoids  con- 
sidered have  straight  lines  joining  corresponding  points  in  the 
two  cross-sections.     The    center  line  must  be  straight  between 
two  cross-sections.     If  a  ridge  or  valley  is  found  lying  diago- 
nally across  the  roadbed,  a  cross-section  must  be  interpolated  at 
the  lowest  (or  highest)  point  of  the  profile.    Therefore  a ' '  break  ' ' 
at  any  section  cannot  be  said  to  run  out  at  the  other  section  on 
the  opposite  side  of  the  center.     It  must  run  out  on  the  same 
side  of  the  center  or  possibly  at  the  center.     Yery  frequently 
complicated  cross-sectioning  may  be  avoided  by  computing  the 
volume,  by  some  special  method,  of  a  mound  or  hollow  when 
the  ground  is  comparatively  regular  except  for  the  irregularity 
referred  to. 

88.  Side-hill  work.     When  the  natural  slope  cuts  the  roadbed 
there  is  a  necessity  for  both  cut  and  fill  at  the  same  cross- section. 
When  this  occurs  the  cross-sections  of  both  cut  and  fill  are  often 
so  nearly  triangular  that  they  may  be  considered  as  such  without 


EARTHWORK. 


101 


great  error,  and  the  volumes  may  be  computed  separately  as 
triangular  prismoids  without  adopting  the  more  elaborate  form 
of  computation  so  necessary  for  complicated  irregular  sections. 
When  the  ground  is  too  irregular  for  this  the  best  plan  is  to 
follow  the  uniform  system.  In  computing  the  cut,  as  in  Fig.  53, 
\ 


the  left  side  would  be  as  usual  ;  there  would  be  a  small  center 
cut  and  an  ordinate  of  zero  at  a  short  distance  to  the  right  of  the 
center.  Then,  ignoring  the  fill,  and  applying  Eq.  61  strictly, 
we  have  two  terms  for  the  left  side,  one  for  the  right,  and  the 
term  involving  J£,  which  will  be  ^Mt  in  this  case,  since  hr  =  0, 
and  the  equation  becomes 


Area  =  ^[xfa  +  y^d  —  hf)  +  xrd  + 
The  area  for  fill  may  also  be  computed  by  a  strict  application 


FIG.  54. 


of  Eq.  61,  but  for  Fig.  54  all  distances  for  the  left  side  are  zero 
and  the  elevation  for  the  first  point  out  is  zero,     d  also  must  be 


OF    THE 

UNIVERSITY 


102 


RAILROAD  CONSTRUCTION. 


89. 


Following  the  rule,   §   81,   literally,  the 


yr(o  - 


v  —  yrhr  —  z 


zr(o  - 


considered  as  zero. 
equation  becomes 

Area(Fill)  =  ±[xrkr 
which  reduces  to 


(Note  that  ov,  Ar>  etc.,  have  different  significations  and 
values  in  this  and  in  the  preceding  paragraphs.)  The  "  terminal 
pyramids  '  '  illustrated  in  Fig.  40  are  instances  of  side-hill  work 
for  very  short  distances.  Since  side-hill  work  always  implies 
'both  cut  and  fill  at  the  same  cross-section,  whenever  either  the 
cut  or  fill  disappears  and  the  earthwork  becomes  wholly  cut  or 
wholly  fill,  that  point  marks  the  end  of  the  '  c  side-hill  work,  '  ' 
and  a  cross-  section  should  be  taken  at  this  point. 

89.  Borrow-pits.  The  cross-sections  of  borrow-pits  will  vary 
not  only  on  account  of  the  undulations  of  the  surface  of  the 


.  55. 


ground,  but  also  on  the  sides,  according  to  whether  they  are 
made  by  widening  a  convenient  cut  (as  illustrated  in  Fig.  55) 
or  simply  by  digging  a  pit.  The  sides  should  always  be  prop- 
erly sloped  and  the  cutting  made  cleanly,  so  as  to  avoid  un- 
sightly roughness.  If  the  slope  ratio  on  the  right-hand  side 
(Fig.  55)  is  6-,  the  area  of  the  triangle  is  -Jsm2.  The  area  of  the 
section  is  %[ug+(g+h)v+(h+j)x  +  (j+k)y+(k+m)z—sm*]. 
If  all  the  horizontal  measurements  were  referred  to  one  side  as 
an  origin,  a  formula  similar  to  Eq.  61  could  readily  be  devel- 
oped, but  little  or  no  advantage  would  be  gained  on  account  of 
any  simplicity  of  computation.  Since  the  exact  volume  of  the 
earth  borrowed  is  frequently  necessary,  the  prismoidal  correc- 


§  90.  EARTHWORK.  103 

tion  should  be  computed ;  and  since  such  a  section  as  Fig.  55 
does  not  even  approximate  to  a  three-level  section,  tliL»  method 
suggested  in  §  82  cannot  be  employed.  It  will  then  be  neces- 
sary to  employ  the  exact  method,  §  83,  by  dividing  the  volume 
into  triangular  prismoids  and  taking  the  summation  of  their 
corrections,  found  according  to  the  general  method  of  §  71. 

80.  Correction  for  curvature.  The  volume  of  a  solid,  gen- 
erated by  revolving  a  plane  area  about  an  axis  lying  in  the 
plane  but  outside  of  the  area,  equals  the  product  of  the  given 
area  times  the  length  of  the  path  of  the  center  of  gravity  of  the 
area.  If  the  centers  of  gravity  of  all  cross- sections  lie  in  the 
center  of  the  road,  where  the  length  of  the  road  is  measured, 
there  is  absolutely  no  necessary  correction  for  curvature.  If  all 
the  cross- sections  in  any  given  length  were  exactly  the  same 
and  therefore  had  the  same  eccentricity,  the  correction  for 
curvature  would  be  very  readily  computed  according  to  the 
above  principle.  But  when  both  the  areas  and  the  eccentrici- 
ties vary  from  point  to  point,  as  is  generally  the  case,  a  theo- 
retically exact  solution  is  quite  complex,  both  in  its  derivation 
and  application.  Suppose,  for  simplicity,  a  curved  section  of 
the  road,  of  uniform  cross- sections  and  with  the  center  of  grav- 
ity of  every  cross-section  at  the  same  distance  e  from  the  center 
line  of  the  road.  The  length  of  the  path  of  the  center  of 
gravity  will  be  to  the  length  of  the  center  line  as  R  ±  e :  It. 
Therefore  we  have  True  vol.  :  nominal  vol.  : :  R  ±  e  :  R. 

R  -\-  e 

.-.  True  vol.  =  IA — ~ —  for  a  volume  of  uniform  area  and 

eccentricity.     For  any   other  area   and  eccentricity  we  have, 

R  +  e' 
similarly,  True  vol.'  =  I  A' — ^ — .     This  shows  that  the  effect 

of  curvature  is  the  same  as  increasing  (or  diminishing)  the  area 
by  a  quantity  depending  on  the  area  and  eccentricity,  the 
increased  (or  diminished)  area  being  found  by  multiplying  the 

actual  area  by  the  ratio  — ~ — .  This  being  independent  of  the 
value  of  Z,  it  is  true  for  infinitesimal  lengths.  If  the  eccen- 


104  RAILROAD  CONSTRUCTION.  §91. 

tricity  is  assumed  to  vary  uniformly  between  two  sections,  the 
equivalent  area  of  a  cross-  section  located  midway  between  the 


two   end  cross-sections    would  be  Am-  —    —  ^  -  .     There- 

H 

fore  the  volume  of   a  solid  which,   when  straight,  would  be 
•(Af  +  ^Am  +  A"},  would  then  become 


True  vol.  = 


Subtracting  the  nominal  volume  (the  true  volume  when  the 
prismoid  is  straight),  the 

Correction  =  ±  ^  [_(A '  +  ZAmy  +  (2Am  +  A")e"'] .     (63) 

Another  demonstration  of  the  same  result  is  given  by  Prof. 
C.  L.  Crandall  in  his  "  Tables  for  the  Computation  of  Kail- 
way  and  other  Earthwork,"  in  which  is  obtained  by  calculus 
methods  the  summation  of  elementary  volumes  having  variable 
areas  with  variable  eccentricities.  The  exact  application  of 
Eq.  (63)  requires  that  Am  be  known,  which  requires  laborious 
computations,  but  no  error  worth  considering  is  involved  if  the 
equation  is  written  approximately 

Curv.  corr.  = -^(A  e' +  A"  e"\    .     .     .     (64) 

which  is  the  equation  generally  used.  The  approximation  con- 
sists in  assuming  that  the  difference  between  A'  and  Am  equals 
the  difference  between  Am  and  A"  but  with  opposite  sign. 
The  error  due  to  the  approximation  is  always  utterly  insig- 
nificant. 

91.  Eccentricity  of  the  center  of  gravity.  The  determina- 
tion of  the  true  positions  of  the  centers  of  gravity  of  a  long 
series  of  irregular  cross- sections  would  be  a  very  laborious 
operation,  but  fortunately  it  is  generally  sufficiently  accurate  to 


§91. 


EARTHWORK 


105 


consider  the  cross-sections  as  three-level  ground,  or,  for  side-hill 
work,  to  be  triangular,  for  the  purpose  of  this  correction.     The 


FIG.  56. 


eccentricity  of  the  cross-section  of  Fig.  56  (including  the  grade 
triangle)  may  be  written 


(a-\-d)xtXi 


, 


r_ 

-(Xl-x'>'  < 


The  side  toward  #r  being  considered  positive  in  the  above 
demonstration,  if  xr  >  x^  e  would  be  negative,  i.e.,  the  center 
of  gravity  would  be  on  the  left  side.  Therefore,  for  three-level 
ground,  the  correction  for  curvature  (see  Eq.  64)  may  be 
written 

Correction  =       C^te'  ~  »/)  +  A"(x{'  -  a?/')]. 


Since  the  approximate  volume  of  the  prismoid  is 
i(A  +  A')  =  l-A'  +  I  A"  =  V  +  V", 

in  which    V  and    V"   represent  the  number  of  cubic  yards 
corresponding  to  the  area  at  each  station,  we  may  write 


Corr.  in  cub.  yds.  =  ~[  F'(a>,  '-  »/)+  V"(x{'- 


(66) 


106  RAILROAD   CONSTRUCTION.  §  91. 

It  should  be  noted  that  the  value  of  <?,  derived  in  Eq.  65,  is 
the  eccentricity  of  the  whole  area  including  the  triangle  under 
the  roadbed.  The  eccentricity  of  the  true  area  is  greater  than 
this  and  equals 

true  area  - 


e  X 


true  area 


The  required  quantity  (A'er  of  Eq.  64)  equals  true  area  X  «,, 
which  equals  (true  area  +  ^ob)  X  e.  Since  the  value  of  e  is  very 
simple,  while  the  value  of  el  would,  in  general,  be  a  complex 
quantity,  it  is  easier  to  use  the  simple  value  of  Eq.  65  and  add 
\ab  to  the  area.  Therefore,  in  the  case  of  three-level  ground 
the  sub  tractive  term  ^ab  (§  78)  should  not  be  subtracted  in 
computing  this  correction.  For  irregular  ground,  when  com- 
puted by  the  method  given  in  §§  81  and  82,  which  does  not 
involve  the  grade  triangle,  a  term  \\ciJb  must  be  added  at  every 
station  when  computing  the  quantities  V  and  V"  for  Eq.  66. 

It  should  be  noted  that  the  factor  1  -=-  312,  which  is 
constant  for  the  length  of  the  curve,  may  be  computed  with  all 
necessary  accuracy  and  without  resorting  to  tables  by  remember- 
ing that 

j}  =  .        5730 

degree  of  curve* 

Since  it  is  useless  to  attempt  the  computation  of  railroad 
earthwork  closer  than  the  nearest  cubic  yard,  it  will  frequently 
be  possible  to  write  out  all  curvature  corrections  by  a  simple 
mental  process  upon  a  mere  inspection  of  the  computation  sheet. 
Eq.  66  shows  that  the  correction  for  each  station  is  of  the  form 


r 
p  —  -.      3B  is  generally  a  large  quantity  —  for  a  6°  curve 

Oil' 

it  is  2865.  (XL  —  xr)  is  generally  small.  It  may  frequently  be 
seen  by  inspection  that  the  product  V(xl  —  a;,.)  is  roughly  twice 
or  three  times  37?,  or  perhaps  less  than  half  of  37?,  so  that  the 
corrective  term  for  that  station  may  be  written  2,  3,  or  0  cubic 
yards,  the  fraction  being  disregarded.  For  much  larger  absolute 


§92. 


EARTHWORK. 


107 


amounts  the  correction  must  be  computed  with  a  correspondingly 
closer  percentage  of  accuracy. 

The  algebraic  sign  of  the  curvature  correction  is  best  deter- 
mined by  noting  that  the  center  of  gravity  of  the  cross-section  is 
on  the  right  or  left  side  of  the  center  according  as  xr  is  greater 
or  less  than  xi,  and  that  the  correction  is  positive  if  the  center  of 
gravity  is  on  the  outside  of  the  curve,  and  negative  if  on  the 
inside. 

It  is  frequently  found  that  xt  is  uniformly  greater  (or  uni- 
formly less)  than  xr  throughout  the  length  of  the  curve.  Then 
the  curvature  correction  for  each  station  is  uniformly  positive  or 
negative.  But  in  irregular  ground  the  center  of  gravity  is  apt 
to  be  irregularly  on  the  outside  or  on  the  inside  of  the  curve, 
and  the  curvature  correction  will  be  correspondingly  positive  or 
negative.  If  the  curve  is  to  the  right,  the  correction  will  be 
positive  or  negative  according  as  (xi  —  xr)  is  positive  or  negative ; 
if  the  curve  is  to  the  left,  the  correction  will  be  positive  or  nega- 
tive according  as  (xr  —  xt)  is  positive  or  negative.  Therefore 
when  computing  curves  to  the  right  use  the  form  (xt  —  a?,.)  in 
Eqs.  66  and  68  ;  when  computing  curves  to  the  left  use  the  form 
(xr  —  XL)  in  these  equations ;  the  algebraic  sign  of  the  correction 
will  then  be  strictly  in  accordance  with  the  results  thus  obtained. 

92.  Center  of  gravity  of  side-hill  sections.     In  computing  the 


FIG.  57. 

correction  for  side-hill  work  the  cross  section  would  be  treated 
as  triangular  unless  the  error  involved  would  evidently  be  too 


108  RAILROAD  CONSTRUCTION.  §  92. 

great  to  be  disregarded.  The  center  of  gravity  of  the  triangle 
lies  on  the  line  joining  the  vertex  with  the  middle  of  the  base 
and  at  -J  of  the  length  of  this  line  from  the  base.  It  is  therefore 
equal  to  the  distance  from  the  center  to  the  foot  of  this  line  plus 
•J  of  its  horizontal  projection.  Therefore 


~b       xr      xl       ~b       xr 

~-  4  "  2  +3  "12+6 

I  XI         Xr 

=  6"  +  3"  "  3 


By  the  same  process  as  that  used  in  §  91  the  correction  equation 
may  be  written 


(x?  -V))]. 


Corr.  in  cub.  yds.  =  -1     V>(g  +  W  -  <*>/)    +  V" L-  +  (^"  -  av")      .  (68) 

0/t[_        \*  /  \*  /_J 

It  should  be  noted  that  since  the  grade  triangle  is  not  used  in 
this  computation  the  volume  of  the  grade  prism  is  not  involved 
in  computing  the  quantities  V  and  V" . 

The  eccentricities  of  cross-sections  in  side-hill  work  are 
never  zero,  and  are  frequently  quite  large.  The  total  volume 
is  generally  quite  small.  It  follows  that  the  correction  for 
curvature  is  generally  a  vastly  larger  proportion  of  the  total 
volume  than  in  ordinary  three-level  or  irregular  sections. 

If  the  triangle  is  wholly  to  one  side  of  the  center,  Eq.  67 
can  still  be  used.  For  example,  to  compute  the  eccentricity  of 
the  triangle  of  fill,  Fig.  57,  denote  the  two  distances  to  the 
slope- stakes  by  yr  and  —  yt  (note  the  minus  sign).  Applying 

Eq.  67  literally  (noting  that  ^-  must  here  be  considered  as  nega- 
tive in  order  to  make  the  notation  consistent)  we  obtain 


§  94.  EARTHWORK.  109 

which  reduces  to 

.     I     -    -.     .     (69) 


As  the  algebraic  signs  tend  to  create  confusion  in  these 
formulae,  it  is  more  simple  to  remember  that  for  a  triangle 
lying  on  loth  sides  of  the  center  e  is  always  numerically  equal 

to  -     £  +  (%i  ~  a?r)    ,  and  for  a  triangle  entirely  on  one  side,  e  is 

O    L-2i  -\ 

numerically  equal  to  —   ty  +  the  numerical  sum  of  the  two  dis- 

tances out].     The  algebraic  sign  of  e  is  readily  determinate  as 
in  §  91. 

93.  Example  of  curvature  correction.     Assume  that  the  fill  in 

§  78  occurred  on  a  6°  curve  to  the  right.     —     =  cu-      The 


quantities  210,  507,  etc.,  represent  the  quantities  F"',  V"  ', 
etc.,  since  they  include  in  each  case  the  61  cubic  yards  due  to 
the  grade  prism.  Then 

V(xt  ~~  xr)         210(22.9  -  8.2)     _  3101.7  _ 

37?  2865  2865 

• 

The  sign  is  plus  since  the  center  of  gravity  of  the  cross-sec- 
tion is  on  the  left  side  of  the  center  and  the  road  curves  to  the 
right,  thus  making  the  true  volume  larger.  For  Sta,  18  the 
correction,  computed  similarly,  is  +  3,  and  the  correction  for 
the  whole  section  is  1  +  3  =  4.  For  Sta.  18  -[-  40  the  cor- 
rection is  computed  as  6  yards.  Therefore,  for  the  40  feet,  the 
correction  is  -^-(3  -f-  6)  =  3.6,  which  is  called  4.  Computing 
the  others  similarly  we  obtain  a  total  correction  of  +  16  cubic 
yards. 

94.  Accuracy  of  earthwork  computations.  The  preceding 
methods  give  the  precise  volume  (except  where  approximations 
are  distinctly  admitted)  of  the  prismoids  which  are  supposed  to 
represent  the  volume  of  the  earthwork.  To  appreciate  the 
accuracy  necessary  in  cross-sectioning  to  obtain  a  given  accuracy 


110  RAILROAD   CONSTRUCTION.  §94. 

in  volume,  consider  that  a  fifteen-foot  length  of  the  cross- section, 
which  is  assumed  to  be  straight,  really  sags  0.1  foot,  so  that  the 
cross-section  is  in  error  by  a  triangle  15  feet  wide  and  0.1  foot 
high.  This  sag  0.1  foot  high  would  hardly  be  detected  by  the 
eye,  but  in  a  length  of  100  feet  in  each  direction  it  would  make 
an  error  of  volume  of  1.4  cubic  yards  in  each  of  the  two  pris- 
moids,  assuming  that  the  sections  at  the  other  ends  were  perfect. 
If  the  cross- sections  at  both  ends  of  a  prismoid  were  in  error  by 
this  same  amount,  the  volume  of  that  prismoid  would  be  in  error 
by  2.8  cubic  yards  if  the  errors  of  area  were  both  plus  or  both 
minus.  If  one  were  plus  and  one  minus,  the  errors  would 
neutralize  each  other,  and  it  is  the  compensating  character  of 
these  errors  which  permits  any  confidence  in  the  results  as 
obtained  by  the  usual  methods  of  cross-sectioning.  It  demon- 
strates the  utter  futility  of  attempting  any  closer  accuracy  than 
the  nearest  cubic  yard.  It  will  thus  be  seen  that  if  an  error 
really  exists  at  any  cross-section  it  involves  the  prismoids  on 
'both  sides  of  the  section,  even  though  all  the  other  cross-sections 
are  perfect.  As  a  further  illustration,  suppose  that  cross-sec- 
tions were  taken  by  the  method  of  slope  angle  and  center  depth 
(§  73),  and  that  a  cross-section,  assumed  as  uniform,  sags  0.4 
foot  in  a  width  of  20  feet.  Assume  an  equal  error  (of  same 
sign)  at  the  other  end  of  a  100-foot  section.  The  error  of 
volume  for  that  one  prismoid  is  38  cubic  yards. 

The  computations  further  assume  that  the  warped  surface, 
passing  through  the  end  sections,  coincides  with  the  surface  of 
the  ground.  Suppose  that  the  cross-sectioning  had  been  done 
with  mathematical  perfection ;  and,  to  assume  a  simple  case, 
suppose  a  sag  of  0.5  foot  between  the  sections,  which  causes  an 
error  equal  to  the  volume  of  a  pyramid  having  a  base  of  20  feet 
(in  each  cross-section)  times  100  feet  (between  the  cross-sections) 
and  a  height  of  0.5  foot.  The  volume  of  this  pyramid  is 
J(20  X  100)  X  0.5  =  333  cub.  ft.  =  12  cub.  yds.  And  yet 
this  sag  or  hump  of  6  inches  would  generally  be  utterly  un- 
noticed, or  at  least  disregarded. 

When  the  ground  is  very  rough  and  broken  it  is  sometimes 


§  96.  EARTHWORK.  Ill 

practically  impossible,  even  with  frequent  cross-sections,  to 
locate  warped  surfaces  which  will  closely  coincide  with  all  the 
sudden  irregularities  of  the  ground.  In  sucli  cases  the  compu- 
tations are  necessarily  more  or  less  approximate  and  dependence 
must  be  placed  on  the  compensating  character  of  the  errors. 

95.  Approximate   computations  from  profiles.     As   a   means 
of  comparing  the  relative   amounts  of  earthwork    on   two  or 
more  proposed  routes  which  have  been  surveyed  by  preliminary 
surveys,  it  will  usually  be  sufficiently  accurate  to  compare  the 
areas  of  cutting  (assuming  that  the  cut  and  fill  are  approximately 
balanced)  as  shown  by  the  several  profiles.      The  errors  involved 
may  be  large  in  individual  cases  and  for  certain  small  sections, 
but  fortunately   the  errors  (in    comparing   two   lines)  will  be 
largely  compensated.     The  errors  are  much  larger  on  side-hill 
work   than  when    the    cross-sections   are   comparatively   level. 
The  errors  become  large  when  the  depth  of  cut  or  fill  is  very 
great.     If  the  lines  compared  have  the  same  general  character 
as  to  the  slope  of  the  cross-sections,  the  proportion  of  side-hill 
work,  and  the  average  depth  of  cut  or  fill,  the  error  involved  in 
considering  their  relative  volumes  of  cutting  to  be  as  the  relative 
areas  of  cutting  on  the  profiles  (obtained  perhaps  by  a  planim- 
eter)  will  probably  be  small.     If  the  volume  in  each  case  is 
computed  by  assuming  the  sections  as  level,  with  a  depth  equal 
to  the  center  cut,  the  error  involved  will  depend  only  on  the 
amount  of  side-hill  work  and  the  degree  of  the  slope.     If  these 
features  are  about  the  same  on  the  two  lines  compared,  the  error 
involved  is  still  less. 

FORMATION    OF    EMBANKMENTS. 

96.  Shrinkage  of  earthwork.     The  evidence  on  this  subject 
as  to  the  amount  of  shrinkage  is  very  conflicting,  a  fact  which 
is  probably  due  to  the  following  causes : 

1.  The  various  kinds  of  earthy  material  act  very  differently 
as  respects  shrinkage.  There  has  been  but  little  uniformity  in 
the  classification  of  earths  in  the  tests  and  experiments  that 
have  been  made. 


112  RAILROAD  CONSTRUCTION.  §  96. 

2.  Very  much  depends  on  the  method  of  forming  an  em- 
bankment (as  will  be  shown  later).     Different  reports  have  been 
based    on   different   methods — often    without    mention    of  the 
method. 

3.  An  embankment  requires  considerable  time  to  shrink  to 
its   final   volume,    and    therefore   much    depends   on   the   time 
elapsed  between  construction  and  the  measurement  of  what  is 
supposed  to  be  the  settled  volume. 

P.  J.  Flynn  quotes  some  experiments  (Eng.  News,  May  1, 
1886)  made  in  India  in  which  pits  were  dug,  having  volumes  of 
400  to  600  cubic  feet.  The  material,  when  piled  into  an  em- 
bankment, measured  largely  in  excess  of  the  original  measure- 
ment— as  is  the  universal  experience.  The  pits  were  refilled 
with  the  same  material.  As  the  rains,  very  heavy  in  India, 
settled  the  material  in  the  pits,  more  was  added  to  keep  the  pits 
full.  Even  after  the  rainy  season  was  over,  there  was  in  every 
case  material  in  excess.  This  would  seem  to  indicate  a  per- 
manent expansion,  although  it  is  possible  that  the  observations 
were  not  continued  for  a  sufficient  time  to  determine  the  final 
settled  volume. 

On  the  contrary,  notes  made  by  Mr.  Elwood  Morris  many 
years  ago  on  the  behavior  of  embankments  of  several  thousand 
cubic  yards,  formed  in  layers  by  carts  and  scrapers,  one  winter 
intervening  between  commencement  and  completion,  showed  in 
each  case  a  permanent  contraction  averaging  about  10$. 

All  authorities  agree  that  rockwork  expands  permanently 
when  formed  into  an  embankment,  but  the  percentages  of 
expansion  given  by  different  authorities  differ  even  more  than 
with  earth — varying  from  8  to  90$.  Of  course  this  very  large 
rang;e  in  the  coefficient  is  due  to  differences  in  the  character  of 
the  rock.  The  softer  the  rock  and  the  closer  its  similarity  to 
earth,  the  less  will  be  its  expansion.  On  account  of  the  conflict- 
ing statements  made,  and  particularly  on  account  of  the  influence 
of  methods  of  work,  but  little  confidence  can  be  felt  in  any 
given  coefficient,  especially  when  given  to  a  fraction  of  a  per 


97. 


EARTHWORK. 


113 


cent,  but  the  consensus  of  American  practice  seems  to  average 
about  as  follows  : 

Permanent  contraction  of  earth  ..........   about  10* 

"         expansion  of  rock  ......  ......  40  to  60* 

These  values  for  rock  should  be  materially  reduced,  according 
to  judgment,  when  the  rock  is  soft  and  liable  to  disintegrate. 
The  hardest  rocks,  loosely  piled,  may  occasionally  give  even 
higher  results.  The  following  is  given  by  several  authors  as 
the  permanent  contraction  of  several  grades  of  earth  : 

Gravel  or  sand  ................  ........  about    8* 

Clay  ...........................  ...  ..       " 

Loam  ..........................  ..;..       " 


12* 


Loose  vegetable  surface  soil 


It  may  be  noticed  from  the  above  table  that  the  harder  and 
cleaner  the  material  the  less  is  the  contraction.  Perfectly  clean 
gravel  or  sand  would  not  probably  change  volume  appreciably. 
The  above  coefficients  of  shrinkage  and  expansion  may  be  used 
to  form  the  following  convenient  table. 


Material. 

To  make  1000  cubic 
yards  of  embankment 
will  require 

1000  cubic  yards  measured 
in  excavation  will  make 

1087  cubic  yards 

920  cubic  yards 

Clay  

1111     " 

900     " 

1136     " 

880     " 

Loose  vegetable  soil        .... 

1176     "           ' 

850     " 

Rock   large  pieces           .... 

714    " 

1400     " 

small      "     

625     " 

1600     " 

measured  in  excavation 

of  embankment. 

97.  Allowance  for  shrinkage.  On  account  of  the  initial 
expansion  and  subsequent  contraction  of  earth,  it  becomes 
necessary  to  form  embankments  higher  than  their  required 
ultimate  form  in  order  to  allow  for  the  subsequent  shrinkage. 
As  the  shrinkage  appears  to  be  all  vertical  (practically),  the 
embankment  must  be  formed  as  shown  in  Fig.  58.  The  effect 


114  RAILROAD  CONSTRUCTION.  §  97. 

of  shrinkage  should  not  be  confounded  with  that  of  slipping  of 
the  sides,  which  is  especially  apt  to  occur  if  the  embankment  is 
subjected  to  heavy  rains  very  soon  after  being  formed,  and  also 
when  the  embankments  are  originally  steep.  It  is  often  difficult 


FIG.  58. 

to  form  an  embankment  at  a  slope  of  1  :  1  which  will  not  slip 
more  or  less  before  it  hardens. 

Yery  high  embankments  shrink  a  greater  percentage  than 
lower  ones.  Various  rules  giving  the  relation  between  shrink- 
age and  height  have  been  suggested,  but  they  vary  as  badly  as 
the  suggested  coefficients  of  contraction,  probably  for  the  same 
causes.  As  the  fact  is  unquestionable,  however,  the  extra 
height  of  the  embankment  must  be  varied  somewhat  as  in  Fig. 
59,  which  represents  a  longitudinal  section  of  an  embankment. 


As  considerable  time  generally  elapses  between  the  completion 
of  the  embankment  and  the  actual  running  of  trains,  the  grade 
ad  will  generally  be  nearly  flattened  down  to  its  ultimate  form 
before  traffic  commences,  but  such  grades  are  occasionally  objec- 
tionable if  added  to  what  is  already  a  ruling  grade.  With  some 
kinds  of  soil  the  time  required  for  complete  settlement  may  be 
as  much  as  two  or  three  years,  but,  even  in  such  cases,  it  is 


§  98.  EARTHWORK.  115 

probable  that  one-half  of  the  settlement  will  take  place  during 
the  first  six  months.  The  engineer  should  therefore  require 
the  contractor  to  make  all  fills  about  8  to  15$  (according  to 
the  material)  higher  than  the  profiles  call  for,  in  order  that 
subsequent  shrinkage  may  not  reduce  it  to  less  than  the  re- 
quired volume. 

98.  Methods  of  forming  embankments.  When  the  method  is 
not  otherwise  objectionable,  a  high  embankment  can  be  formed 
very  cheaply  (assuming  that  carts  or  wheelbarrows  are  used)  by 
dumping  over  the  end  and  building  to  the  full  height  (or  even 
higher,  to  allow  for  shrinkage)  as  the  embankment  proceeds. 
This  allows  more  time  for  shrinkage,  saves  nearly  all  the  cost  of 
spreading  (see  Item  4,  §  111),  and  reduces  the  cost  of  roadways 
(Item  5).  Of  course  this  method  is  especially  applicable  when 
the  material  comes  from  a  place  as  high  as  or  higher  than  grade, 
so  that  no  up-hill  hauling  is  required. 

Another  method  is  to  spread  it  in  layers  two  or  three  feet 
thick  (see  Fig.  60),  which  are  made  concave  upwards  to  avoid 


W///////////W//7^ 

FIG.  60. 

possible  sliding  on  each  other.  Spreading  in  layers  has  the 
advantage  of  partially  ramming  each  layer,  so  that  the  subse- 
quent shrinkage  is  very  small.  Sometimes  small  trenches  are 
dug  along  the  lines  of  the  toes  of  the  embankment.  This  will 
frequently  prevent  the  sliding  of  a  large  mass  of  the  embank- 
ment, which  will  then  require  extensive  and  costly  repairs,  to 
say  nothing  of  possible  accidents  if  the  sliding  occurs  after  the 
road  is  in  operation.  Incidentally  these  trenches  will  be  of 
value  in  draining  the  subsoil.  When  circumstances  require  an 
embankment  on  a  hillside,  it  is  advisable  to  cut  out  "  steps  "  to 
prevent  a  possible  sliding  of  the  whole  embankment.  Merely 


116  RAILROAD   CONSTRUCTION.  §  99. 

ploughing  the  side-hill  will  often  be  a  cheaper  and  sufficiently 
effective  method. 


FIG.  61. 


Occasionally  the  formation  of  a  very  high  and  long  embank- 
ment may  be  most  easily  and  cheaply  accomplished  by  building 
a  trestle  to  grade  and  opening  the  road.  Earth  can  then  be 
procured  where  most  convenient,  perhaps  several  miles  away, 
loaded  on  cars  with  a  steam-shovel,  hauled  by  the  trainload,  and 
dumped  from  the  cars  with  a  patent  unloader.  On  such  a  large 
scale,  the  cost  per  yard  would  be  very  much  less  than  by  ordi- 
nary methods — enough  less  sometimes  to  more  than  pay  for  the 
temporary  trestle,  besides  allowing  the  road  to  be  opened  for 
traffic  very  much  earlier,  which  is  often  a  matter  of  prime 
financial  importance.  It  may  also  obviate  the  necessity  for 
extensive  borrow-pits  in  the  immediate  neighborhood  of  the 
heavy  fill  and  also  iitilize  material  which  would  otherwise  be 
wasted. 

COMPUTATION    OF    HAUL. 

99.  Nature  of  subject,  As  will  be  shown  later  when  analyz- 
ing the  cost  of  earthwork,  the  most  variable  item  of  cost  is  that 
depending  on  the  distance  hauled.  As  it  is  manifestly  imprac- 
ticable to  calculate  the  exact  distance  to  which  every  individual 
cartload  of  earth  has  been  moved,  it  becomes  necessary  to  devise 
a  means  which  will  give  at  least  an  equivalent  of  the  haulage  of 
all  the  earth  moved.  Evidently  the  average  haul  for  any  mass 
of  earth  moved  is  equal  to  the  distance  from  the  center  of  grav- 
ity of  the  excavation  to  the  center  of  gravity  of  the  embank- 


§  100. 


EARTHWORK. 


117 


ment  formed  by  the  excavated  material.  As  a  rough  approxi- 
mation the  center  of  gravity  of  a  cut  (or  fill)  may  sometimes  be 
considered  to  coincide  with  the  center  of  gravity  of  that  part  of 
the  profile  representing  it,  but  the  error  is  frequently  very  large. 
The  center  of  gravity  may  be  determined  by  various  methods, 
but  the  method  of  the  "  mass  diagram  "  accomplishes  the  same 
ultimate  purpose  (the  determination  of  the  haul)  with  all-suffi- 
cient accuracy  and  also  furnishes  other  valuable  information. 

100.  Mass  diagram.     In  Fig.  62  let  A ' B'  .  .  .  G'  represent 
a  profile  and  grade  line  drawn  to  the  usual  scales.     Assume  A' 


FIG.  62.— MASS  DIAGRAM. 


to  be  a  point  past  which  no  earthwork  will  be  hauled.  Above 
every  station  point  in  the  profile  draw  an  ordinate  which 
will  represent  the  algebraic  sum  of  the  cubic  yards  of  cut  and 
fill  (calling  cut  +  and  fill  —  )  from  the  point  A  to  the  point 
considered.  In  doing  this  shrinkage  must  be  allowed  for  by 
considering  how  much  embankment  would  actually  be  made  by 
so  many  cubic  yards  of  excavation  of  such  material.  For 
example,  it  will  be  found  that  1000  cubic  yards  of  sand  or 
gravel,  measured  in  place  (see  §  97),  will  make  about  920  cubic 
yards  of  embankment  ;  therefore  all  cuttings  in  sand  or  gravel 
should  be  discounted  in  about  this  proportion.  Excavations  in 
rock  should  be  increased  in  the  proper  ratio.  In  short,  all  ex- 
cavations should  be  valued  according  to  the  amount  of  settled 
embankment  that  could  be  made  from  them.  The  computations 
may  be  made  systematically  as  shown  in  the  tabular  form.  Place 


118 


RAILROAD   CONSTRUCTION. 


101. 


in  the  first  column  a  list  of  the  stations;  in  the  second  column, 
the  number  of  cubic  yards  of  cut  or  fill  between  each  station 
and  the  preceding  station ;  in  the  third  and  fourth  columns,  the 
kind  of  material  and  the  proper  shrinkage  factor ;  in  the  fifth 
column,  a  repetition  of  the  quantities  in  cubic  yards,  except  that 
the  excavations  are  diminished  (or  increased,  in  the  case  of  rock) 
to  the  number  of  cubic  yards  of  settled  embankment  which  may 
be  made  from  them.  In  the  sixth  column,  place  the  algebraic 
sum  of  the  quantities  in  the  fifth  column  (calling  cuts  +  and 
tills  — )  from  the  starting-point  to  the  station  considered.  These 
algebraic  sums  at  each  station  will  be  the  ordinates,  drawn  to 
some  scale,  of  the  mass  curve.  The  scale  to  be  used  will  depend 
somewhat  on  whether  the  work  is  heavy  or  light,  but  for  ordi- 
nary cases  a  scale  of  5000  cubic  yards  per  inch  may  be  used. 
Drawing  these  ordinates  to  scale,  a  curve  A.,  I>,  .  .  .  G  may  be 
obtained  by  joining  the  extremities  of  the  ordinates. 


Sta. 

Yards]  Sf  + 

Material. 

Shrinkage 
factor. 

Yards,  reduced 
for  shrinkage. 

Ordinate  in 
mass  curve. 

40  +  70 

o 

47 
48 
+  60 
49 

+  195 
+  1793 
+  614 
-  143 

Clayey  soil 

—  10  per  cent 
-10 

-10   " 

+  175 
+  1613 
+  553 
—  143 

+  175 

+  1788 
+  2341 

+  2198 

50 

—  906 

—  906 

+  1292 

51 

—  1985 

—  1985 

693 

52 

—  1731 

—  1731 

—  2414 

-|-30 

-  113 

—  112 

—  2526 

53 

+  70 
54 

+  177 
+  180 
-  53 

Hard  rock 
<  <    « 

+60  per  cent 
+60 

+  283 
+  289 
—  -52 

-2243 
—  1954 
—  2006 

+  42 

—  71 

71 

2077 

55 
56 

57 

+  376 
+  1342 
+  1302 

Clayey  soil 

«        X 

—  10  percent 
-10 
-10 

+  249 
+  1118 
+  1172 

-1828 
-  710 
+  462 

101.  Properties  of  the  mass  curve. 

1.  The  curve  will  be  rising  while   over  cuts   and  falling 
while  over  fills. 

2.  A  tangent  to  the  curve  will  be  horizontal  (as  at  B,  D,  E, 
F,  and  G)  when  passing  from  cut  to  fill  or  from  fill  to  cut. 


§  101.  EARTHWORK.  119 

3.  When  the  curve  is  below  the  "  zero  line  "  it  shows  that 
material  must  be  drawn  backward  (to  the  left) ;  and  vice  versa, 
when  the  curve  is  above  the  zero  line  it  shows  that  material 
must  be  drawn  forward  (to  the  right). 

4.  When  the  curve  crosses  the  zero  line  (as  at  A  and  C)  it 
shows  (in  this  instance)  that  the  cut  between  A'  and  B'  will  just 
provide  the  material  required  for  the  fill  bet  ween  B'  and  C',  and 
that  no  material  should  be  hauled  past  C',  or,  in  general,  past 
any  intersection  of  the  mass  curve  and  the  zero  line. 

o.   If  any  horizontal  line  be  drawn  (as  ab),  it  indicates  that 
the  cut  and  fill  between  a'  and  b'  will  just  balance. 

6.  When    the    center   of    gravity    of    a   given    volume   of 
material  is  to  be  moved  a  given  distance,  it  makes  no  difference 
(at  least  theoretically)  how  far  each  individual  load  may  be 
hauled  or  how  any  individual  load  may  be  disposed  of.     The 
summation   of  the  products   of   each  load   times   the  distance 
hauled  will  be  a  constant,  whatever  the  method,  and  will  equal 
the  total  volume  times  the  movement  of  the  center  of  gravity. 
The  average  haul,  which  is  the  movement  of  the  center  of 
gravity,  will  therefore  equal  the  summation  of  these  products 
divided  by  the  total  volume.     If  we  draw  two  horizontal  par- 
allel lines   at  an  infinitesimal  distance  dx  apart,  as  at  ab,  the 
small  increment  of  cut  dx  at  a'  will  fill  the  corresponding  incre- 
ment of  fill  at  b',  and  this  material  must  be  hauled  the  distance 
ab.     Therefore  the  product  of  ab  and  dx,  which  is  the  product 
of  distance  times  volume,  is  represented  by  the  area  of  the 
infinitesimal  rectangle  at  ab,  and  the  total  area  ABC  represents 
the  summation  of  volume  times  distance  for  all  the  earth  move- 
ment between  A'  and  C' .     This  summation  of  products  divided 
by  the  total  volume  gives  the  average  haul. 

7.  The  horizontal  line,  tangent  at  E  and  cutting  the  curve 
at  e,f,  and  g,  shows  that  the  cut  and  fill  between  e'  and  E'  will 
just   balance,  and  that  a  possible  method  of  hauling  (whether 
desirable  or  not)    would  be  to  "borrow"  earth   for   the   fill 
between  C'  and  e',  use  the  material  between  D'  and  E'  for  the 


120  RAILROAD   CONSTRUCTION.  §  101. 

fill  between  e  and  D',  and  similarly  balance  cut  and  fill  between 
E'  and/'  and  also  between/'  and  g' . 

8.  Similarly  the  horizontal  line  hldrn  may  be  drawn  cutting 
the  curve,  which  will  show  another  possible  method  of  hauling. 
According  to  this  plan,  the  fill  between  G'  and  K'  would  be 
made  by .  borrowing ;  the  cut  and  fill  between  h'  and  Jc'  would 
balance;  also  that  between  k'  and  I'  and  between  I'  and  in'. 
Smce  Ihe  area  ehDkE  represents  the  measure  of  haul  for  the 
earth  between  e  and  E' ',  and  the  other  areas  measure  the  corre- 
sponding hauls  similarly,  it  is  evident  that  the  sum  of  the  areas 
ehDkE  and  ElFmf,  which  is  the  measure  of  haul  of  all  the 
material  between  e'  and  /',  is  largely  in  excess  of  the  sum  of 
the  areas  hDk,  kEl,  and  IFm,  plus  the  somewhat  uncertain 
measures  of  haul  due  to  borrowing  material  for  e'h'  and  wasting 
the  material  between  in  and/*'.  Therefore  to  make  the  meas- 
ure of  haul  a  minimum  a  line  should  be  drawn  which  will 
make  the  sum  of  the  areas  between  it  and  the  mass  curve  a 
minimum.  Of  course  this  is  not  necessarily  the  cheapest  plan, 
as  it  implies  more  or  less  borrowing  and  wasting  of  material, 
which  may  cost  more  than  the  amount  saved  in  haul.  The 
comparison  of  the  two  methods  is  quite  simple,  however.  Since 
the  amount  of  fill  between  e'  and  h'  is  represented  by  the  differ- 
ence of  the  ordinates  at  e  and  A,  and  similarly  for  m'  and/',  it 
follows  that  the  amount  to  be  borrowed  between  e'  and  //  will 
exactly  equal  the  amount  wasted  between  mf  and  /'.  By  the 
first  of  the  above  methods  the  haul  is  excessive,  but  is  definitely 
known  from  the  mass  diagram,  and  all  of  the  material  is  util- 
ized ;  by  the  second  method  the  haul  is  reduced  to  about  one- 
half,  but  there  is  a  known  quantity  in  cubic  yards  wasted  at  one 
place  and  the  same  quantity  borrowed  at  another.  The  length 
of  haul  necessary  for  the  borrowed  material  would  need  to  be 
ascertained ;  also  the  haul  necessary  to  waste  the  other  material 
at  a  place  where  it  would  be  unobjectionable.  Frequently  this 
is  best  done  by  widening  an  embankment  beyond  its  necessary 
width.  The  computation  of  the  relative  cost  of  the  above 
methods  will  be  discussed  later  (§  116). 


§  102.  EARTHWORK.  121 

9.  Suppose  that  it  were  deemed  best,  after  drawing  the  mass 
curve,  to  introduce  a  trestle  between  s'  and  v\  thus  saving  an 
amount  in  till  equal  to  tv.  If  such  had  been  the  original  design, 
the  mass  curve  would  have  been  a  straight  horizontal  line 
between  s  and  t  and  would  continue  as  a  curve  which  would  be 
at  all  points  a  distance  tv  above  the  curve  vFtnzfGg.  If  the 
line  Ef  is  to  be  used  as  a  zero  line,  its  intersection  with  the  new 
curve  at  x  will  show  that  the  material  between  E'  and  z'  will 
just  balance  if  the  trestle  is  used,  and  that  the  amount  of  haul 
will  be  measured  by  the  area  between  the  line  Ex  and  the  broken 
line  Estx.  The  same  computed  result  may  be  obtained  without 
drawing  the  auxiliary  curve  txn  ...  by  drawing  the  horizontal 
line  zy  at  a  distance  xz  (=  tv)  below  Ex.  The  amount  of  the 
haul  can  then  be  obtained  by  adding  the  triangular  area  between 
Es  and  the  horizontal  line  Ex,  the  rectangle  between  st  and  Ex, 
and  the  irregular  area  between  vFz  and  y  .  .  .  z  (which  last  is 
evidently  equal  to  the  area  between  tx  and  E .  .  .  x).  The  dis- 
posal of  the  material  at  the  right  of  z'  would  then  be  governed 
by  the  indications  of  the  profile  and  mass  diagram  which  would 
be  found  at  the  right  of  g '.  In  fact  it  is  difficult  to  decide  with 
the  best  of  judgment  as  to  the  proper  disposal  of  material  with- 
out having  a  mass  diagram  extending  to  a  considerable  distance 
each  side  of  that  part  of  the  road  under  immediate  considera- 
tion. 

102.  Area  of  the  mass  curve.  The  area  may  be  computed 
most  readily  by  means  of  a  planimeter,  which  is  capable  with 
reasonable  care  of  measuring  such  areas  with  as  great  accuracy 
as  is  necessary  for  this  work.  If  no  such  instrument  is  obtain- 
able, the  area  may  be  obtained  by  an  application  of  "  Simpson's 
rule."  The  orclinates  will  usually  be  spaced  100  feet  apart. 
Select  an  even  number  of  such  spaces,  leaving,  if  necessary,  one 
or  more  triangles  or  trapezoids  at  the  ends  for  separate  and 
independent  computation.  Let  y0  .  .  .  yn  be  the  ordinates,  i.e., 
the  number  of  cubic  yards  at  each  station  of  the  mass  curve,  or 
the  figures  of  ''column  six"  referred  to  in  §  100.  Let  the 
uniform  distance  between  ordinates  (=100  feet)  be  called  1,  i.e., 


122  RAILROAD  CONSTRUCTION.  §  103. 

one  station.     Then  the  units  of  the  resulting  area  will  be  cubic 
yards  hauled  one  station.     Then  the 

Area  =  *[>.  +  4(yi  +  2/3  +  ...  y(H  _  1})  +  2(ya  +  y*  +  •  •  •  y(n_8))  +  yn  ]•      (70) 

"When  an  ordinate  occurs  at  a  substation,  the  best  plan  is  to 
ignore  it  at  first  and  calculate  the  area  as  above.  Then,  if  the 
difference  involved  is  too  great  to  be  neglected,  calculate  the 
area  of  the  triangle  having  the  extremity  of  the  ordinate  at  the 
substation  as  an  apex,  and  the  extremities  of  the  ordinates  at  the 
adjacent  stations  as  the  ends  of  the  base.  This  may  be  done  by 
finding  the  ordinate  at  the  substation  that  would  be  a  propor- 
tional between  the  ordinates  at  the  adjacent  full  stations.  Sub- 
tract this  from  the  real  ordinate  (or  vice  versa)  and  multiply  the 
difference  by  i  X  1.  An  inspection  will  often  show  that  the 
correction  thus  obtained  would  be  too  small  to  be  worthy  of  con- 
sideration. If  there  is  more  than  one  substation  between  two 
full  stations,  the  corrective  area  will  consist  of  two  triangles  and 
one  or  more  trapezoids  which  may  be  similarly  computed,  if 
necessary. 

When  the  zero  line  (Fig.  62)  is  shifted  to  eE^  the  drop  from 
AC  (produced)  to  E  is  known  in  the  same  units,  cubic  yards. 
This  constant  may  be  subtracted  from  the  numbers  ("  column 
4,"  §  100)  representing  the  ordinates,  and  will  thus  give,  with- 
out any  scaling  from  the  diagram,  the  exact  value  of  the  modi- 
fied ordinates. 

103.  Value  of  the  mass  diagram.  The  great  value  of  the  mass 
diagram  lies  in  the  readiness  with  which  different  plans  for  the 
disposal  of  material  may  be  examined  and  compared.  When 
the  mass  curve  is  once  drawn,  it  will  generally  require  only  a 
shifting  of  the  horizontal  line  to  show  the  disposal  of  the  material 
by  any  proposed  method.  The  mass  diagram  also  shows  the 
extreme  length  of  haul  that  will  be  required  by  any  proposed 
method  of  disposal  of  material.  This  brings  into  consideration 
the  "  limit  of  profitable  haul,"  which  will  be  fully  discussed  in 
§  116.  For  the  present  it  may  be  said  that  with  each  method 
of  carrying  material  there  is  some  limit  beyond  which  the  expense 


§  104.  EARTHWORK.  123 

of  hauling  will  exceed  the  loss  resulting  from  borrowing  and 
wasting.  With  wheelbarrows  and  scrapers  the  limit  of  profit- 
able haul  is  comparatively  short,  with  carts  and  train-cars  it  is 
much  longer,  while  with  locomotives  and  cars  it  may  be  several 
miles.  If,  in  Fig.  62,  eE  or  Ef  exceeds  the  limit  of  profitable 
haul,  it  shows  at  once  that  some  such  line  as  Jiklm  should  be 
drawn  and  the  material  disposed  of  accordingly. 

104.  Changing  the  grade  line.  The  formation  of  the  mass 
curve  and  the  resulting  plans  as  to  the  disposal  of  material  are 
based  on  the  mutual  relations  of  the  grade  line  and  the  surface 
profile  and  the  amounts  of  cut  and  fill  which  are  thereby  im- 
plied. If  the  grade  line  is  altered,  every  cross-section  is 
altered,  the  amount  of  cut  and  fill  is  altered,  and  the  mass 
curve  is  also  changed.  At  the  farther  limit  of  the  actual 
change  of  the  grade  line  the  revised  mass  curve  will  have  (in 
general)  a  different  ordinate  from  the  previous  ordinate  at  that 
point.  From  that  point  on,  the  revised  mass  curve  will  be  par- 
allel to  its  former  position,  and  the  revised  curve  may  be  treated 
similarly  to  the  case  previously  mentioned  in  which  a  trestle  was 
introduced.  Since  it  involves  tedious  calculations  to  determine 
accurately  how  much  the  volume  of  earthwork  is  altered  by  a 
change  in  grade  line,  especially  through  irregular  country,  the 
effect  on  the  mass  curve  of  a  change  in  the  grade  line  cannot 
therefore  be  readily  determined  except  in  an  approximate  way. 
Raising  the  grade  line  will  evidently  increase  the  fills  and 
diminish  the  cuts,  and  vice  versa.  Therefore  if  the  mass  curve 
indicated,  for  example,  either  an  excessively  long  haul  or  the 
necessity  for  borrowing  material  (implying  a  fill)  and  wasting 
material  farther  on  (implying  a  cut),  it  would  be  possible  to 
diminish  the  fill  (and  hence  the  amount  of  material  to  be  bor- 
rowed) by  lowering  the  grade  line  near  that  place,  and  diminish 
the  cut  (and  hence  the  amount  of  material  to  be  wasted)  by 
raising  the  grade  line  at  or  near  the  place  farther  on.  Whether 
the  advantage  thus  gained  would  compensate  for  the  possibly 
injurious  effect  of  these  changes  on  the  grade  line  would  require 
patient  investigation.  But  the  method  outlined  shows  how  the 


124 


RAILROAD   CONSTRUCTION. 


105. 


mass  curve  might  be  used  to  indicate  a  possible  change  in  grade 
line  which  might  be  demonstrated  to  be  profitable. 

105.  Limit  of  free  haul.  It  is  sometimes  specified  in  con- 
tracts for  earthwork  that  all  material  shall  be  entitled  to  free 
haul  up  to  some  specified  limit,  say  500  or  1000  feet,  and  that 
all  material  drawn  farther  than  that  shall  be  entitled  to  an 
allowance  on  the  excess  of  distance.  It  is  manifestly  imprac- 
ticable to  measure  the  excess  for  each  load,  as  much  so  as  to 
measure  the  actual  haul  of  each  load.  The  mass  diagram  also 
solves  this  problem  very  readily.  Let  Fig.  63  represent  a  pro- 


FIG.  03. 

file  and  mass  diagram  of  about  2000  feet  of  road,  and  suppose 
that  800  feet  is  taken  as  the  limit  of  free  haul.  Find  two 
points,  a  and  5,  in  the  mass  curve  which  are  on  the  same  hori- 
zontal line  and  which  are  800  feet  apart.  Project  these  points 
down  to  a'  and  ~b '.  Then  the  cut  and  fill  between  a  and  U  will 
just  balance,  and  the  cut  between  A  and  a'  will  be  needed  for 
the  fill  between  b'  and  C'.  In  the  mass  curve,  the  area  between 
the  horizontal  line  ab  and  the  curve  aB~b  represents  the  haulage 
of  the  material  between  a'  and  £',  which  is  all  free.  The  rect- 
angle abmn  represents  the  haulage  of  the  material  in  the  cut 
A' a'  across  the  800  feet  from  a'  to  l> .  This  is  also  free.  The 
sum  of  the  two  areas  Aam  and  InC  represents  the  haulage 
entitled  to  an  allowance,  since  it  is  the  summation  of  the  products 
of  cubic  yards  times  the  excess  of  distance  hauled. 


§  105.  EARTHWORK.  125 

If  the  amount  of  cut  and  fill  was  symmetrical  about  the 
point  B',  the  mass  curve  would  be  a  symmetrical  curve  about  the 
vertical  line  through  Z?,  and  the  two  limiting  lines  of  free  haul 
would  be  placed  symmetrically  about  B  and  Bf .  In  general 
there  is  no  such  symmetry,  and  frequently  the  difference  is  con- 
siderable. The  area  aBbnm  will  be  materially  changed  accord- 
ing as  the  two  vertical  lines  am  and  bn,  always  800  feet  apart, 
are  shifted  to  the  right  or  left.  It  is  easy  to  show  that  the  area 
aBbnm  is  a  maximum  when  ab  is  horizontal.  The  minimum 
value  would  be  obtained  either  when  m  reached  A  or  n  reached 
C,  depending  on  the  exact  form  of  the  curve.  Since  the  posi- 
tion for  the  minimum  value  is  manifestly  unfair,  the  best  definite 
value  obtainable  is  the  maximum,  which  must  be  obtained  as 
above  described.  Since  aBbnm  is  made  maximum,  the  re- 
mainder of  the  area,  which  is  the  allowance  for  overhaul,  be- 
comes a  minimum.  The  areas  Aam  and  bCn  may  be  obtained 
as  in  §  102.  If  the  whole  area  AaBbCA  has  been  previously 
computed,  it  may  be  more  convenient  to  compute  the  area 
aBbnm  and  subtract  it  from  the  total  area. 

Since  the  intersections  of  the  mass  curve  and  the  "  zero  line  " 
mark  limits  past  which  no  material  is  drawn,  it  follows  that 
there  will  be  no  allowance  for  overhaul  except  where  the  dis- 
tance between  consecutive  intersections  of  the  zero  line  and  mass 
curve  exceeds  the  limit  of  free  haul. 

Frequently  all  allowances  for  overhaul  are  disregarded ;  the 
profiles,  estimates  of  quantities,  and  the  required  disposal  of  ma- 
terial are  shown  to  bidding  contractors,  and  they  must  then  make 
their  own  allowances  and  bid  accordingly.  This  method  has 
the  advantage  of  avoiding  possible  disputes  as  to  the  amount  of 
the  overhaul  allowance,  and  is  popular  with  railroad  companies  on 
this  account.  On  the  other  hand  the  facility  with  which  differ- 
ent plans  for  the  disposal  of  material  may  be  studied  and  com- 
pared by  the  mass- curve  method  facilitates  the  adoption  of  the 
most  economical  plan,  and  the  elimination  of  uncertainty  will 
frequently  lead  to  a  safe  reduction  of  the  bid,  and  so  the  method 
is  valuable  to  both  the  railroad  company  and  the  contractor. 


126  RAILROAD   CONSTRUCTION.  §  106. 


ELEMENTS    OF    THE    COST    OF    EARTHWORK. 

(The  following  analysis  of  the  cost  of  earthwork  follows  the 
general  method  given  in  the  well-known  papers  published  by 
Ellwood  Morris,  C.E.,  in  the  Journal  of  the  Franklin  Institute 
in  September  and  October,  1841.  Numerous  corroborative 
data  have  been  obtained  from  various  other  sources,  and  also 
figures  on  methods  not  then  in  vogue.) 

106.  General  divisions  of  the  subject.  The  variations  in  the 
cost  of  earthwork  are  caused  by  the  greatly  varying  conditions 
under  which  the  work  is  done,  chief  among  which  is  character 
of  material,  method  of  carriage,  and  length  of  haul.  Any  gen- 
eral system  of  computation  must  therefore  differentiate  the  total 
cost  into  such  elementary  items  that  all  differences  due  to  varia- 
tions in  conditions  may  be  allowed  for.  The  variations  due  to 
character  of  material  will  be  allowed  for  by  an  estimate  on  loose 
light  sandy  soil,  and  also  an  estimate  on  the  heaviest  soils,  such 
as  stiff  clay  and  hard-pan.  These  represent  the  extremes  (ex- 
cluding rock,  which  will  be  treated  separately),  and  the  cost  of 
intermediate  grades  must  be  estimated  by  interpolating  between 
the  extreme  values.  The  general  divisions  of  the  subject  will 
be:* 

1.  Loosening. 

2.  Loading. 

3.  Hauling. 

4.  Spreading. 

5.  Keeping  roadways  in  order. 

6.  Repairs,  wear,  depreciation,  and  interest  on  cost  of  plant. 

7.  Superintendence  and  incidentals. 

8.  Contractor's  profit. 

By  making  the  estimates  on  the  basis  of  $1  per  day  for  the 
cost  of  common  labor,  it  is  a  simple  matter  to  revise  the  esti- 
mates according  to  the  local  price  of  labor  by  multiplying  the 
final  estimate  of  cost  by  the  price  of  labor  in  dollars  per  day. 

*  Trautwine. 


§  107.  EARTHWORK.  127 

107.  Item  1.  LOOSENING,  (a)  Ploughs.  Very  light  sandy 
soils  can  frequently  be  shovelled  without  any  previous  loosening, 
but  it  is  generally  economical,  even  with  very  light  material,  to 
use  a  plough.  Morris  quotes,  as  the  results  of  experiments, 
that  a  three-horse  plough  would  loosen  from  250  to  800  cubic 
yards  of  earth  per  day,  which  at  a  valuation  of  $5  per  day 
would  make  the  cost  per  yard  vary  from  2  cents  to  0.6  cent. 
Trautwine  estimates  the  cost  on  the  basis  of  two  men  handling 
a  two-horse  plough  at  a  total  cost  of  $3.87  per  day,  being  $1 
each  for  the  men,  75  c.  for  each  horse,  and  an  allowance  of  37  c. 
for  the  plough,  harness,  etc.  From  200  to  600  cubic  yards  is 
estimated  as  a  fair  day's  work,  which  makes  a  cost  of  1.9c.  to 
0.65c.  per  yard,  which  is  substantially  the  same  estimate  as 
above.  Extremely  heavy  soils  have  sometimes  been  loosened 
by  means  of  special  ploughs  operated  by  traction-engines. 

(b)  Picks.     When  picks  are  used  for  loosening  the  earth,  as 
is  frequently  necessary  and  as  is  often  done  when  ploughing 
would  perhaps  be  really  cheaper,  an  estimate  *  for  a  fair  day's 
work  is  from  14  to  60  cubic  yards,  the  14  yards  being  the  esti- 
mate for  stiff  clay  or  cemented  gravel,  and  the  60  yards  the  esti- 
mate for  the  lightest  soil  that  would  require  loosening.     At  81 
per  day  this  means  about  7  c.  to  1.7  c.  per  cubic  yard,  which  is 
about  three  times  the  cost  of  ploughing.     Five  feet  of  the  face 
is  estimated  f  as  the  least  width  along  the  face  of  a  bank  that 
should  be  allowed  to  enable  each  laborer  to  work  with  freedom 
and  hence  economically. 

(c)  Blasting.     Although  some  of  the  softer  shaly  rocks  may 
be  loosened  with  a  pick  for  about  15  to  20  c.  per  yard,  yet  rock 
in  general,  frozen  earth,  and  sometimes  even  compact  clay  is 
most  economically  loosened  by  blasting.     The  subject  of  blast- 
ing will  be  taken  up  later,  §§  117-123. 

(d)  Steam-shovels.    The  items  of  loosening  and  loading  merge 
together  with  this  method,  which  will  therefore  be  treated  in 
the  next  section. 

*  Trautwine.  t  Hurst. 


128  RAILROAD  CONSTRUCTION.  §  108. 

108,  Item  2.  LOADING,  (a)  Hand- shovelling.  Much  depends 
on  proper  management,  so  that  the  shovellers  need  not  wait 
unduly  either  for  material  or  carts.  With  the  best  of  manage- 
ment considerable  time  is  thus  lost,  and  yet  the  intervals  of  rest 
need  not  be  considered  as  entirely  lost,  as  it  enables  the  men  to 
work,  while  actually  loading,  at  a  rate  which  it  would  be  physi- 
cally impossible  for  them  to  maintain  for  ten  hours.  Seven 
shovellers  are  sometimes  allowed  for  each  cart ;  otherwise  there 
should  be  five,  two  on  each  side  and  one  in  the  rear.  Economy 
requires  that  the  number  of  loads  per  cart  per  day  should  be 
made  as  large  as  possible,  and  it  is  therefore  wise  to  employ  as 
many  shovellers  as  can  work  without  mutual  interference  and 
without  wasting  time  in  waiting  for  material  or  carts.  The 
figures  obtainable  for  the  cost  of  this  item  are  unsatisfactory  on 
account  of  their  large  disagreements.  The  following  are  quoted 
as  the  number  of  cubic  yards  that  can  be  loaded  into  a  cart  by 
an  average  laborer  in  a  working  day  of  ten  hours,  the  lower 
estimate  referring  to  heavy  soils,  and  the  higher  to  light  sandy 
soils :  10  to  14  cubic  yards  (Morris),  12  to  17  cubic  yards  (Has- 
koll),  18  to  22  cubic  yards  (Hurst),  17  to  24  cubic  yards  (Traut- 
wine),  16  to  48  cubic  yards  (Ancelin).  As  these  estimates  are 
generally  claimed  to  be  based  on  actual  experience,  the  discre- 
pancies are  probably  due  to  differences  of  management.  If  the 
average  of  15  to  25  cubic  yards  be  accepted,  it  means,  on  the 
basis  of  $1  per  day,  6.7c.  to  4c.  per  cubic  yard.  These  esti- 
mates apply  only  to  earth.  Rockwork  costs  more,  not  only 
because  it  is  harder  to  handle,  but  because  a  cubic  yard  of  solid 
rock,  measured  in  place,  occupies  about  1.8  cubic  yards  when 
broken  up,  while  a  cubic  yard  of  earth  will  occupy  about  1.2 
cubic  yards.  Rock  work  will  therefore  require  about  50$  more 
loads  to  haul  a  given  volume,  measured  in  place,  than  will  the 
same  nominal  volume  of  earthwork.  The  above  authorities  give 
estimates  for  loading  rock  varying  from  6.9c.  to  10  c.  per  cubic 
yard.  The  above  estimates  apply  only  to  the  loading  of  carts 
or  cars  with  shovels  or  by  hand  (loading  masses  of  rock).  The 


§  108.  EARTH WORK.  129 

cost  of  loading  wheelbarrows  and  the  cost  of  scraper  work  will 
be  treated  under  the  item  of  hauling. 

(b)  Steam- shovels.*  Whenever  the  magnitude  of  the  work 
will  warrant  it  there  is  great  economy  in  the  use  of  steam-shovels. 
These  have  a  "  bucket"  or  "  dipper"  on  the  end  of  a  long 
beam,  the  bucket  having  a  capacity  varying  from  J  to  2£  cubic 
yards.  Steam-shovels  handle  all  kinds  of  material  from  the 
softest  earth  to  shale  rock,  earthy  material  containing  large 
boulders,  tree-stumps,  etc.  The  capacity  of  the  larger  sizes  is 
about  3000  cubic  yards  in  10  hours.  They  perform  all  the 
work  of  loosening  and  loading.  Their  economical  working 
requires  that  the  material  shall  be  hauled  away  as  fast  as  it  can 
be  loaded,  which  usually  means  that  cars  on  a  track,  hauled  by 
horses  or  mules,  or  still  better  by  a  locomotive,  shall  be  used. 
The  expenses  for  a  steam-shovel,  costing  about  $5000,  will 
average  about  $1000  per  month.  Of  this  the  engineer  will  get 
$100 ;  the  fireman  $50 ;  the  cranesman  $90 ;  repairs  perhaps 
$250  to  $300;  coal,  from  15  to  25  tons,  cost  very  variable  on 
account  of  expensive  hauling;  water,  a  very  uncertain  amount, 
sometimes  costing  $100  per  month;  about  five  laborers  and  a 
foreman,  the  laborers  getting  $1.25  per  day  and  the  foreman 
$2.50  per  day,  which  will  amount  to  $227.50  per  month. 
This  gang  of  laborers  is  employed  in  shifting  the  shovel  when 
necessary,  taking  up  and  relaying  tracks  for  the  cars,  shifting 
loaded  and  unloaded  cars,  etc.  In  shovelling  through  a  deep 
cut,  the  shovel  is  operated  so  as  to  undermine  the  upper  parts 
of  the  cut,  which  then  fall  down  within  reach  of  the  shovel,  thus 
increasing  the  amount  of  material  handled  for  each  new  position 
of  the  shovel.  If  the  material  is  too  tough  to  fall  down  by  its 
own  weight,  it  is  sometimes  found  economical  to  employ  a  gang 
of  men  to  loosen  it  or  even  blast  it  rather  than  shift  the  shovel 
so  frequently.  Non-condensing  engines  of  50  horse-power  use 
so  much  water  that  the  cost  of  water-supply  becomes  a  serious 

*  For  a  thorough  treatment  of  the  capabilities,  cost,  and  management  of 
rteain-shovels  the  reader  is  referred  to  "Steam-shovels  and  Steam-shovel 
Work,"  by  E.  A.  Hermann.  D.  Van  Nostrand  Co.,  New  York. 


130  RAILROAD  CONSTRUCTION.  §  109. 

matter  if  water  is  not  readily  obtainable.  The  lack  of  water 
facilities  will  often  justify  the  construction  of  a  pipe  line  from 
some  distant  source  and  the  installation  of  a  steam-pump. 
Hence  the  seemingly  large  estimate  of  $100  per  month  for 
water-supply,  although  under  favorable  circumstances  the  cost 
may  almost  vanish.  The  larger  steam-shovels  will  consume 
nearly  a  ton  of  coal  per  day  of  10  hours.  The  expense  of  haul- 
ing this  coal  from  the  nearest  railroad  or  canal  to  the  location  of 
the  cut  is  often  a  very  serious  item  of  expense  and  may  easily 
double  the  cost  per  ton.  Some  steam-shovels  have  been  con- 
structed to  be  operated  by  electricity  obtained  from  a  plant 
perhaps  several  miles  away.  Such  a  method  is  especially 
advantageous  when  fuel  and  water  are  difficult  to  obtain. 

109.  Item  3.  HAULING.  The  cost  of  hauling  depends  on  the 
number  of  round  trips  per  day  that  can  be  made  by  each  vehicle 
employed.  As  the  cost  of  each  vehicle  is  practically  the  same 
whether  it  makes  many  trips  or  few,  it  becomes  important  that 
the  number  of  trips  should  be  made  a  maximum,  and  to  that 
end  there  should  be  as  little  delay  as  possible  in  loading  and  un- 
loading. Therefore  devices  for  facilitating  the  passage  of  the 
vehicles  have  a  real  money  value. 

(a)  Carts.  The  average  speed  of  a  horse  hauling  a  two- 
wheeled  cart  has  been  found  to  be  200  feet  per  minute,  a  little 
slower  when  hauling  the  load  and  a  little  faster  when  returning 
empty.  This  figure  has  been  repeatedly  verified.  It  means  an 
allowance  of  one  minute  for  each  100  feet  (or  "station")  of 
"lead — the  lead  being  the  distance  the  earth  is  hauled."  The 
time  lost  in  loading,  dumping,  waiting  to  load,  etc.,  has  been 
found  to  average  4  minutes  per  load.  Representing  the  num- 
ber of  stations  (100  feet)  of  lead  by  s,  the  number  of  loads 
handled  in  10  hours  (600  minutes)  would  be  600  ~  (s  +  4).  The 
number  of  loads  per  cubic  yard,  measured  in  the  bank,  is  differ- 
entiated by  Morris  into  three  classes,  viz.  : 

3  loads  per  cubic  yard  in  descending  hauling ; 
3J  "       "       "        "      "  level  hauling;   and 

4  "       "       "        u     u  ascending  hauling. 


§  109.  EARTHWORK.  131 

Attempts  have  been  made  to  estimate  the  effect  of  the  grade 
of  the  roadway  by  a  theoretical  consideration  of  its  rate,  and  of 
the  comparative  strength  of  a  horse  on  a  level  and  on  various 
grades.  While  such  computations  are  always  practicable  on  a 
railway  (even  on  a  temporary  construction  track),  the  traction  on 
a  temporary  earth  roadway  is  always  very  large  and  so  very 
variable  that  any  refinements  are  useless.  On  railroad  earth- 
work the  hauling  is  generally  nearly  level  or  it  is  descending  — 
forming  embankments  on  low  ground  with  material  from  cuts  in 
high  ground.  The  only  common  exception  occurs  when  an 
embankment  is  formed  from  borrow-pits  on  low  ground.  One 
method  of  allowing  for  ascending  grade  is  to  add  to  the  hori- 
zontal distance  14  times  the  difference  of  elevation  for  work 
with  carts  and  24  times  the  difference  of  elevation  for  work 
with  wheelbarrows,  and  use  that  as  the  lead.  For  example, 
using  carts,  if  the  lead  is  300  feet  and  there  is  a  difference  of 
elevation  of  20  feet,  the  lead  would  be  considered  equivalent  to 
300  +  (14  X  20)  =  580  feet  on  a  level. 

Trautwirie  assumes  the  average  load  for  all  classes  of  work 
to  be  £  cubic  yard,  which  figure  is  justified  by  large  experience. 
Using  one  figure  for  all  classes  of  work  simplifies  the  calculations 
and  gives  the  number  of  cubic  yards  carried  per  day  of  10  hours 


equal  to  57  —    -pr.     Dividing  the  cost  of  a  cart  per  day  by  the 

6(S  4~  *J 

number  of  cubic  yards  carried  gives  the  cost  of  hauling  per 
yard.  In  computing  the  cost  of  a  cart  per  day,  Trau  twine 
refers  to  the  practice  of  having  one  driver  manage  four  carts, 
thus  making  a  charge  of  25  c.  per  day  for  each  cart  for  the 
driver.  75  c.  is  allowed  for  the  horse,  which  is  supposed  to  be 
the  total  cost,  including  that  for  Sundays  and  rainy  days.  25  c. 
more  is  allowed  for  the  cart,  harness,  repairs,  etc.  ,  thus  making 
a  total  cost  of  $1.25  per  day.  Some  contractors  employ  a 
greater  number  of  drivers  and  expect  each  to  assist  in  loading. 
There  is  found  to  be  no  saving  in  total  cost  per  yard,  while  the 
chances  of  loafing  are  perhaps  greater.  Morris  instances  five 
actual  cases  in  which  the  cost  of  the  cart  (reduced  to  the  basis  of 


132  RAILROAD  CONSTRUCTION.  §  109. 

$1  per  day  for  labor)  varied  from  $1.37  to  $1.48.     The  items 
of  these  costs  were  not  given. 

Since  the  time  required  for  loading  loose  rock  is  greater  than 
for  earthwork,  less  loads  will  be  hauled  per  day.  The  time 
allowance  for  loading,  etc.,  is  estimated  by  Trautwine  as  6 
minutes  instead  of  4  as  for  earth.  Considering  the  great  ex- 
pansion of  rock  when  broken  up  (see  §  97),  one  cubic  yard  of 
solid  rock,  measured  in  place,  would  furnish  the  equivalent  of 
five  loads  of  earthwork  of  -J  cubic  yard.  Therefore,  on  the 
basis  of  five  loads  per  cubic  yard,  the  number  of  cubic  yards. 

handled  per  day  per  cart  would  be    ,     .    „». 

o(s  -j-  o) 


Cost  per  yard  in  cents  =  -  '  — .      .     (71) 


(b)  Wagons.     For  longer  leads   (i.e.,  from  J-  to  f  of  a  mile) 
wagons  drawn  by  two  horses  have  been  found  most  economical. 
The  wagons  have  bottoms  of  loose  thick  narrow  boards  and  are 
unloaded  very  easily  and  quickly  by  lifting  the  individual  boards 
and  breaking  up  the  continuity  of  the  bottom,  thus  depositing 
the  load  directly  underneath  the  wagon.     The  capacity  is  about 
one  cubic  -yard.     The  cost  may  be  estimated  on  the  same  prin- 
ciples as  that  for  carts. 

(c)  Wheelbarrows.     According  to  Trautwine,    the    speed    of 
moving  wheelbarrows  may  be  considered  the  same  as  for  carts, 
200  feet  per  minute  ;  the  time  spent  in  loading  and  dumping  is 
1£  minutes,  and  in  addition  about  TL  of  the  time  is  wasted  in 
short  rests,  adjusting  the  wheeling  plants,  etc.      On  the  basis  of 
$1  per  day  for  labor,  an  allowance  of  5  c.  for  the  barrow,  and  14 
loads  per  cubic  yard,  the  cost  of  hauling  per  cubic  yard  (com- 
puted on  the  same  principles  as  above)  will  be 


105  XU(«  +  1.25) 
600  X  0.9 


§  109.  EARTHWORK.  133 

For  rockwork  the  number  of  loads  per  cubic  yard  is  estimated 
as  24,  and  the  time  spent  in  loading,  etc.,  estimated  at  1.6  min- 
utes instead  of  1.25  minutes,  which  makes  the  estimate 


Cost  per  cubic  yard  =  +  ^>.       .     (73) 


(d)  Scrapers.  *  Scrapers,  or  scoops,  are  especially  useful  in 
canal  work,  and  also  for  railroad  work  when  a  low  embankment 
is  to  be  formed  from  borrow-pits  at  the  sides,  when  the  distance 
does  not  exceed  100  feet,  nor  the  vertical  height  15  feet.  The 
slope  should  not  exceed  1 . 5  to  1 .  Under  these  conditions  scraper 
work  is  cheaper  than  any  other  method.  Scooping  may  be  done 
all  in  one  direction,  in  which  case  two  half -turns  are  made  for 
each  load  moved ;  or  it  may  be  done  in  both  directions  (from 
both  sides  on  to  a  bank,  or,  in  canal  work,  from  the  center  to 
each  bank),  in  which  case  one  load  is  hauled  to  each  half -turn. 
The  capacity  of  the  scoops  (the  u  drag  "  variety)  is  -JL  cubic 
yard ;  the  time  lost  in  loading,  unloading,  and  all  other  ways 
per  load  (except  in  turning)  will  average  f  minute ;  the  time  lost 
in  each  half- turn  (semi-circle)  is  J  minute ;  the  speed  of  the 
horses  may  be  estimated  as  70  feet  of  lead  per  minute,  the  lead 
being  here  considered  as  the  sum  of  the  vertical  and  horizontal 
distances,  and  the  estimate  including  the  time  of  going  and  re- 
turning. If  a  represents  the  sum  of  the  horizontal  and  vertical 
distances,  the  number  of  cubic  yards  handled  per  day  of  10 
hours  by  "  side- scooping"  will  be 

4200 
ich  equals  - 

For  u  double-scooping  "  the  formula  becomes 

/     600    \ 
•/  -  -  \  4200 

°-1l  i-  +  i  )'  whlch  e(iuals  7qry0' 
\        / 

*  Condensed  from  Journ.  Franklin  Inst.,  Oct.  1841,  by  Morris. 


134  RAILROAD   CONSTRUCTION.  §  109. 

Dividing  the  cost  of  a  scraper  per  day  (estimated  at  $2.75)  by 
the  number  of  yards  handled  per  day  gives  the  average  cost  per 
yard. 

Except  in  very  loose  sandy  soil  it  is  best  to  plough  the  earth 
first,  which  will  cost  about  1  c.  per  yard.  (See  §  107.)  Drag- 
scrapers  are  now  made  chiefly  of  steel,  and  their  capacity  is  more 
nearly  0.15  cubic  yard.  Wheeled  scrapers,  having  a  capacity 
of  about  0.5  cubic  yard,  are  frequently  used  with  even  greater 
economy  and  for  greater  distances,  as  they  are  cheaper  than 
carts  up  to  250  or  300  feet  of  lead.  Both  drag-  and  wheel- 
scrapers  are  best  operated  in  gangs  of  perhaps  10,  using  extra 
or  "  snap  "  teams  to  help  load,  and  a  few  extra  men  to  help  in 
loading  and  unloading.  The  average  cost  of  one  scraper  per 
day  may  thus  be  easily  calculated  and  the  average  number  of 
cubic  yards  handled  per  day  computed  as  above,  from  which 
the  cost  per  yard  may  be  estimated. 

(e)  Cars  and  horses.  The  items  of  cost  by  this  method  are 
{a)  charge  for  horses  employed,  (b)  charge  for  men  employed 
strictly  in  hauling,  (<?)  charge  for  shifting  rails  when  necessary, 
(d)  repairs,  depreciation,  and  interest  on  cost  of  cars  and  track. 
Part  of  this  cost  should  strictly  be  classified  under  items  5  and 
6,  mentioned  in  §  106,  but  it  is  perhaps  more  convenient  to 
estimate  them  as  follows. 

The  traction  of  a  car  on  rails  is  so  very  small  and  constant 
that  grade  resistance  constitutes  a  very  large  part  of  the  total 
resistance  if  the  grade  is  \%  or  more.  For  all  ordinary  grades 
it  is  sufficiently  accurate  to  say  that  the  grade  resistance  is  to 
the  gross  weight  as  the  rise  is  to  the  distance.  If  the  distance 
is  supposed  to  be  measured  along  the  slope,  the  proportion  is 
strictly  true;  i.e.,  on  a  \<f>  grade  the  grade  resistance  is  1  Ib. 
per  100  of  weight  or  20  Ibs.  per  ton.  If  the  resistance  on  a 
level  at  the  usual  velocity  is  TJ^,  a  grade  of  1 :  120  (0.83$)  will 
exactly  double  it.  If  the  material  is  hauled  down  a  grade  of 
1 :  120,  the  cars  will  run  by  gravity  after  being  started.  The 
work  of  hauling  will  then  consist  practically  of  hauling  the 
empty  cars  up  the  grade.  The  grade  resistance  depends  only 


§  109.  EARTHWORK.  135 

on  the  rate  of  grade  and  the  weight,  but  the  tractive  resistance 
will  be  greater  per  ton  of  weight  for  the  unloaded  than  for  the 
loaded  cars.  The  tractive  power  of  a  horse  is  less  on  a  grade 
than  on  a  level,  not  only  because  the  horse  raises  his  own  weight 
in  addition  to  the  load,  but  is  anatomically  less  capable  of 
pulling  on  a  grade  than  on  a  level.  In  general  it  will  be  pos- 
sible to  plan  the  work  so  that  loaded  cars  need  not  be  hauled  up 
a  grade,  unless  an  embankment  is  to  be  formed  from  a  low 
borrow-pit,  in  which  case  another  method  would  probably  be 
advisable.  These  computations  are  chiefly  utilized  in  designing 
the  method  of  work — the  proportion  of  horses  to  cars.  An 
example  may  be  quoted  from  English  practice  (Hurst),  in  which 
the  cars  had  a  capacity  of  3J  cubic  yards,  weighing  30  cwt. 
empty.  Two  horses  took  five  "  wagons  "  f  of  a  mile  on  a  level 
railroad  and  made  15  journeys  per  day  of  10  hours,  i.e.,  they 
handled  250  yards  per  day.  In  addition  to  those  on  the 
"straight  road,"  another  horse  was  employed  to  make  up  the 
train  of  loaded  wagons.  With  a  short  lead  the  straight-road 
horses  were  employed  for  this  purpose.  In  the  above  example 
the  number  of  men  required  to  handle  these  cars,  shift  the 
tracks,  etc.,  is  not  given,  and  so  the  exact  cost  of  the  above 
work  cannot  be  analyzed.  It  may  be  noticed  that  the  two 
horses  travelled  22  j-  miles  per  day,  drawing  in  one  direction  a 
load,  including  the  weight  of  the  cars,  of  about  57,300  Ibs.  or 
28.65  net  tons.  Allowing  T^-  as  the  necessary  tractive  force, 
it  would  require  a  pull  of  477.5  Ibs.,  or  239  Ibs.  for  each  horse. 
With  a  velocity  of  220  feet  per  minute  this  would  amount  to 
1£  horse-power  per  horse,  exerted  for  only  a  short  time,  how- 
ever, and  allowing  considerable  time  for  rest  and  for  drawing 
only  the  empty  cars.  The  cars  generally  used  in  this  country 
have  a  capacity  of  1J  cubic  yards  and  cost  about  §65  apiece. 
Besides  the  shovellers  and  dumping-gang,  several  men  and  a 
foreman  will  be  required  to  keep  the  track  in  order  and  to  make 
the  constant  shifts  that  are  necessary.  Two  trains  are  generally 
used,  one  of  which  is  loaded  while  the  other  is  run  to  the 
dump.  Some  passing-place  is  necessary,  but  this  is  generally 


186  RAILROAD  CONSTRUCTION.  §  109. 

provided  by  having  a  switch  at  the  cut  and  running  the  trains 
on  eacli  track  alternately.  This  insures  a  train  of  cars  always 
at  the  cut  to  keep  the  shovellers  employed.  The  cost  of  haul- 
ing per  cubic  yard  can  only  be  computed  when  the  number  of 
laborers,  cars,  and  horses  employed  are  known,  and  these  will 
depend  on  the  lead,  on  the  character  of  the  excavation,  on  the 
grade,  if  any,  etc. ,  and  must  be  so  proportioned  that  the  shovel- 
lers need  not  wait  for  cars  to  fill,  nor  the  dumping-gang  for 
material  to  handle,  nor  the  horses  and  drivers  for  cars  to  haul. 
Much  skill  is  necessary  to  keep  a  large  force  in  smooth  running 
order. 

(f )  Cars  and  locomotives.  30-lb.  rails  are  the  lightest  that 
should  be  used  for  this  work,  and  35-  or  40-lb.  rails  are  better. 
One  or  two  narrow-gauge  locomotives  (depending  on  the  length 
of  haul),  costing  about  $2500  each,  will  be  necessary  to  handle 
two  trains  of  about  15  cars  each,  the  cars  having  a  capacity  of 
about  2  cubic  yards  and  costing  about  $100  each.  Some  cars 
can  be  obtained  as  low  as  $70.  A  force  of  about  five  men  and 
a  foreman  will  be  required  to  shift  the  tracks.  The  track- 
shifters,  except  the  foreman,  may  be  common  laborers.  The 
dumping-gang  will  require  about  seven  men.  Even  when  the 
material  is  all  taken  down  grade  the  grades  may  be  too  steep  for 
the  safe  hauling  of  loaded  cars  down  the  grade,  or  for  hauling 
empty  cars  up  the  grade.  Under  such  circumstances  temporary 
trestles  are  necessary  to  reduce  the  grade.  When  these  are 
used,  the  uprights  and  bracing  are  left  in  the  embankment — 
only  the  stringers  being  removed.  This  is  largely  a  necessity, 
but  is  partially  compensated  by  the  fact  that  the  trestle  forms  a 
core  to  the  embankment  which  prevents  lateral  shifting  during 
settlement.  The  average  speed  of  the  trains  may  be  taken  as 
10  miles  per  hour  or  5  miles  of  lead  per  hour.  The  time  lost 
in  loading  and  unloading  is  estimated  (Trautwine)  as  9  minutes 
or  .15  of  an  hour.  The  number  of  trips  per  day  of  10  hours 

.,,          ,  10 50 

|  (miles  of  lead)  +  .15  °r  (miles  of  lead)  +  .75' 
course  this  quotient  must  be  a  whole  number.     Knowing  the 


§110.  EARTHWORK.  137 

number  of  trains  and  their  capacity,  the  total  number  of  cubic 
yards  handled  is  known,  which,  divided  into  the  total  daily  cost 
of  the  trains,  will  give  the  cost  of  hauling  per  yard.  The  daily 
cost  of  a  train  will  include 

(a)  Wages  of  engineer,  who  frequently  fires  his  own  engine ; 

(I)  Fuel,  about  J  to  1  ton  of  bituminous  coal,  depending  on 
work  done; 

(c)  Water,  a  very  variable  item,  frequently  costing  $3  to  $5 
per  day ; 

(d)  Repairs,  variable,  frequently  at  rate  of  50  to  60$  per 
year; 

(e)  Interest  on  cost  and  depreciation,  16  to  40$. 

To  these  must  be  added,  to  obtain  the  total  cost  of  the  haul, 

(f)  Wages  of  the  gang  employed  in  shifting  track. 

110.  Choice  of  method  of  haul  dependent  on  distance, 
In  light  side-hill  work  in  which  material  need  not  be  moved 
more  than  12  or  15  feet,  i.e.,  moved  laterally  across  the  road- 
bed, the  earth  may  be  moved  most  cheaply  by  mere  shovelling. 
Beyond  12  feet  scrapers  are  more  economical.  At  about  100 
feet  drag-scrapers  and  wheelbarrows  are  equally  economical. 
Between  100  and  200  feet  wheelbarrows  are  generally  cheaper 
than  either  carts  or  drag-scrapers,  but  wheeled  scrapers  are 
always  cheaper  than  wheelbarrows.  Beyond  500  feet  two- 
wheeled  carts  become  the  most  economical  up  to  about  1700 
feet ;  then  four-wheeled  wagons  become  more  economical  up  to 
3500  feet.  Beyond  this  cars  on  rails,  drawn  by  horses  or  by 
locomotives,  become  cheaper.  The  economy  of  cars  on  rails 
becomes  evident  for  distances  as  small  as  300  feet  provided  the 
volume  of  the  excavation  will  justify  the  outlay.  Locomotives 
will  always  be  cheaper  than  horses  and  mules  providing  the 
work  to  be  done  is  of  sufficient  magnitude  to  justify  the  pur- 
chase of  the  necessary  plant  and  risk  the  loss  in  selling  the  plant 
ultimately  as  second-hand  equipment,  or  keeping  the  plant  on 
hand  and  idle  for  an  indefinite  period  waiting  for  other  work. 
Horses  will  not  be  economical  for  distances  much  over  a  mile. 
For  greater  distances  locomotives  are  more  economical,  but  the 


138  RAILROAD   CONSTRUCTION.  §111- 

question  of  "limit  of  profitable  haul"  (§  116)  must  be  closely 
studied,  as  the  circumstances  are  certainly  not  common  when  it 
is  advisable  to  haul  material  much  over  a  mile. 

111.  Item  4.     SPREADING.    The  cost  of  spreading  varies  with 
the  method  employed  in  dumping  the  load.     When  the  earth  is 
tipped   over  the  edge  of  an  embankment  there  is  little  if  any 
necessary  work.     Trautwine  allows  about  J  c.  per  cubic  yard 
for  keeping  the  dumping-places  clear  and  in  order.     This  would 
represent  the  wages  of  one  man  at  $1  per  day  attending  to  the 
unloading  of  1 2 OO  two- wheeled  carts  each  carrying  £  cubic  yard. 
1200    carts   in    10    hours  would   mean  an  average  of  two  per 
minute,  which  implies  more  rapid  and  efficient  work  than  may  be 
depended  on.     The  allowance  is  probably  too  small.     When  the 
material   is    dumped  in  layers  some  levelling  is  required,   for 
which  Trautwine  allows  50  to  100  cubic  yards  as  a  fair  day's 
work,  costing  from   1  to  2  cents  per  cubic  yard.     The  cost  of 
spreading  will  not  ordinarily  exceed  this  and  is  frequently  noth- 
ing— all  depending  on  the  method  of  unloading.      It  should  be 
noted  that  Mr.  Morris's  examples  and  computations  (Jour.  Frank- 
lin Inst.,  Sept.  1841)  disregard  altogether  any  special  charge 
for  this  item. 

112.  Item  5.    KEEPING  ROADWAYS  IN  ORDER.    This  feature 
is  important  as  a  measure  of  true  economy,  whatever  the  system 
of  transportation,  but  it  is  often  neglected.      A  petty  saving  in 
such  matters  will  cost  many  times  as  much  in  increased  labor  in 
hauling  and  loss  of  time.     With  some  methods  of  haul  the  cost 
is  best  combined  with  that  of  other  items. 

(a)  Wheelbarrows.  Wheelbarrows  should  generally  be  run 
on  planks  laid  on  the  ground.  The  adjusting  and  shifting  of 
these  planks  is  done  by  the  wheelers,  and  the  time  for  it  is  allowed 
for  in  the  10$  allowance  for  "short  rests,  adjusting  the  wheel- 
ing plank,  etc."  The  actual  cost  of  the  planks  must  be  added, 
but  it  would  evidently  be  a  very  small  addition  per  cubic  yard 
in  a  large  contract.  When  the  wheelbarrows  are  run  on  planks 
placed  on  ' '  horses  "  or  on  trestles  the  cost  is  very  appreciable ; 
but  the  method  is  frequently  used  with  great  economy.  The 


§  114.  EARTHWORK.  139 

variations  in  the  requirements  render  any  general  estimate  of 
such  cost  impracticable. 

(b)  Carts  and  wagons.     The   cost    of   keeping   roadways   in 
order  for  carts  and  wagons  is  sometimes  estimated  merely  as  so 
much  per  cubic  yard,  but  it  is  evidently  a  function  of  the  lead. 
The  work  consists  in  draining  off  puddles,  filling  up  ruts,  pick- 
ing up  loose  stones  that   may  have  fallen  off  the  loads,  and  in 
general  doing  everything  that  will  reduce  the  traction  as  much 
as  possible.     Temporary  inclines,  built  to  avoid  excessive  grade 
at  some  one  point,  are  often  measures  of  true  economy.     Traut- 
wine  suggests  yL  c.  per  cubic  yard  per  100  feet  of  lead  for  earth- 
work and  -f$  c.  for  rockwork,  as  an  estimate  for  this  item  when 
carts  are  used. 

(c)  Cars.      When  cars  are  used  a  shifting-gang,  consisting 
of  a  foreman  and  several  men  (say  five),  are  constantly  employed 
in  shifting  the  track  so  that  the  material  may  be  loaded  and  un- 
loaded where  it  is  desired.     The  aVerage  cost  of  this  item  may 
be  estimated  by  dividing  the  total  daily  cost  of  this  gang  by  the 
number  of  cubic  yards  handled  in  one  day. 

113.  Item  6.    REPAIRS,  WEAR,  DEPRECIATION,  AND  INTEREST 
ON  COST  OF  PLANT.     The  amount  of  this  item  evidently  depends 
upon  the  character  of  the  soil — the  harder  the  soil  the  worse  the 
wear  and  depreciation.     The  interest  on  cost  depends  on  the 
current  borrowing  value  of  money.      The  estimate  for  this  item 
has  already  been  included  in  the  allowances  for  horses,   carts, 
ploughs,  harness,  wheelbarrows,  steam-shovels,  etc.     Trautwine 
estimates  J  c.  per  cubic  yard  for  picks  and  shovels.     Deprecia- 
tion is  generally  a  large  percentage  of  the  cost  of  earth-working 
tools,  the  life  of  all  being  limited  to  a  few  years,  and  of  many 
tools  to  a  few  months. 

114.  Item  7.    SUPERINTENDENCE  AND  INCIDENTALS.    The  inci- 
dentals include  water-carriers,  trimming  cuts  to  grade,  digging 
the  side  ditches,  trimming  up  the  sides  of  borrow-pits  to  prevent 
their  becoming  unsightly,  etc.     These  last  operations  yield  but 
little  earth  and  cost  far  more  than  the  price  paid  per  cubic  yard. 
Morris  allows  1  c.  per   cubic  yard   for  this    item ;     Trautwine 


140  RAILROAD  CONSTRUCTION. 

allows  If  to  2  c.  for  it;  while  others  combine  items  6  and  7 
and  call  them  5$  of  the  total  cost,  which  method  has  the  merit 
of  making  the  cost  of  items  6  and  7  a  function  of  the  character 
of  soil  and  length  of  lead. 

115,  ItemS,   CONTRACTOR'S  PROFIT.    This  is  usually  estimated 
at  from  6  to  15$,  according  to  the  sharpness  of  the  competition 
and  the  possible  uncertainty  as  to  true  cost  owing  to  unfavorable 
circumstances.    The  contractor's  real  profit  may  vary  considerably 
from  this.     He  often  pays  clerks,  boards  and  lodges  the  laborers 
in  shanties  built  for  the  purpose,  or  keeps  a  supply-store,  and 
has  various  other  items  both  of  profit  and  expense.     His  profit 
is  largely  dependent  on  skill  in  so  handling  the  men  that  all  can 
work  effectively  without  interference  or  delays  in  waiting  for 
others.     An  unusual  season  of  bad  weather  will  often  affect  the 
cost  very  seriously.     It  is  a  common  occurrence  to  find  that  two 
contractors  may  be  working  on  the  same  kind  of  material  and 
under  precisely  similar  conditions  and  at  the  same  price,  and  yet 
one  may  be  making  money  and  the  other  losing  it — all  on  ac- 
count of  difference  of  management. 

116.  Limit  of  profitable  haul.     As  intimated  in  §§   103  and 
110,  there  is  with  every  method  of  haul  a  limit  of  distance  be- 
yond which  the  expense  for  excessive  hauling  will  exceed  the 
loss  resulting  from  borrowing  and  wasting.      This  distance  is 
somewhat  dependent  on  local  conditions,  thus  requiring  an  inde- 
pendent solution  for  each  particular  case,  but  the  general  prin- 
ciples involved  will  be  about  as  follows :  Assume  that  it  has  been 
determined,  as  in  Fig.  62,  that  the  cut  and  fill  will  exactly  bal- 
ance between  two  points,  as  between  e  and  a?,  assuming  that,  as 
indicated  in  §  101  (9),  a  trestle  has  been  introduced  between  s 
and  t,  thus  altering  the  mass  curve  to  Estxn  .  .  .     Since  there 
is  a  balance  between  A'  and  C' ',  the  material  for  the  fill  between 
C'  and  e'  must  be  obtained  either  by  "  borrowing  "  in  the  im- 
mediate neighborhood  or  by  transportation  from  the  excavation 
between  z    and  ri '.     If  cut  and  fill  have  been  approximately 
balanced  in  the  selection  of  grade  line,  as  is  ordinarily  done, 
borrowing  material  for  the  fill  C'e'  implies  a  wastage  of  material 


£116.  EARTH WORK.  141 

at  the  cut  z'n'.  To  compare  the  two  methods,  we  may  place 
airainst  the  plan  of  borrowing  and  wasting,  (a)  cost,  if  any,  of 
extra  right  of  way  that  may  be  needed  from  which  to  obtain 
earth  for  the  fill  O'e'-,  (b)  cost  of  loosening,  loading,  hauling 
a  distance  equal  to  that  between  the  centers  of  gravity  of  the 
borrow- pit  and  of  the  fill,  and  the  other  expenses  incidental  to 
borrowing  M  cubic  yards  for  the  fill  C'e' ;  (c)  cost  of  loosening, 
loading,  hauling  a  distance  equal  to  that  between  the  centers 
of  gravity  of  the  cut  z'n  and  of  the  spoil-bank,  and  the  other 
expenses  incidental  to  wasting  M  cubic  yards  at  the  cut  z'n' ; 
(d)  cost,  if  any,  of  land  needed  for  the  spoil-bank.  The  cost  of 
the  other  plan  will  be  the  cost  of  loosening,  loading,  hauling  (the 
hauling  being  represented  by  the  trapezoidal  figure  Cexn),  and 
the  other  expenses  incidental  to  making  the  fill  C'e  with  the 
material  from  the  cut  z'n'  ^  the  amount  of  material  being  M  cubic 
yards,  which  is  represented  in  the  figure  by  the  vertical  ordi- 
nate  from  e  to  the  line  On.  The  difference  between  these  costs 
will  be  the  cost,  if  any,  of  land  for  borrow-pit  and  spoil-bank 
plus  the  cost  of  loosening,  loading,  etc.  (except  hauling  and 
roadways)  of  M  cubic  yards,  minus  the  difference  in  cost  of  the 
excessive  haul  from  Ce  to  xn  and  the  comparatively  short  hauls 
from  borrow-pit  and  to  spoil-bank. 

As  an  illustration,  taking  some  of  the  estimates  previously 
given  for  operating  with  average  material,  the  cost  of  all  items, 
except  hauling  and  roadways,  would  be  about  as  follows : 
loosening,  with  plough,  1.2  c.,  loading  5.0  c.,  spreading  1.5  c., 
wear,  depreciation,  etc.,  .25  c.,  superintendence,  etc.,  1.5  c. ; 
total  8.95  c.  Suppose  that  the  haul  for  both  borrowing  and 
wasting  averages  100  feet  or  1  station.  Then  the  cost  of  haul 
per  yard,  using  carts,  would  be  (§  109,  a)  [125  X  3(1  +  4)]  -j- 
600  —  3.125  c.  The  cost  of  roadways  would  be  about  0.1  c. 
per  yard,  making  a  total  of  3.225  c.  per  cubic  yard.  Assume 
M  =  10000  cubic  yards  and  the  area  Cexn  =  180000  yards- 
stations  or  the  equivalent  of  10000  yards  hauled  1800  feet. 
This  haul  would  cost  [125  X  3(18  +  4)]  -r-  600  =  13.75  c.  per 
cubic  yard.  The  cost  of  roadways  will  be  18  X  .1  or  1.8  c., 


142  RAILROAD  CONSTRUCTION.  §  117. 

making  a  total  of  15.55  c.  for  hauling  and  roadways.  The 
difference  of  cost  of  hauling  and  roadways  will  be  15.55  — 
(2  X  3.225)  =  9.10  c.  per  yard  or  $910  for  the  10000  yards. 
Offsetting  this  is  the  cost  of  loosening,  etc.,  10000  yards,  at 
8.95  c.,  costing  $895.  These  figures  may  be  better  compared 
as  follows : 

f  Loosening,  etc.,  10000  yards,  @    8.95  c.  $  895. 

J   Hauling,       "      10000     "       @  15.55  c.                        1555. 
LONG  HAUL,    -j  

$2450. 


Loosening,  etc.,  10000  yards  (borrowed),  @  8.95    c.  $895. 
"      10000      "     (wasted),      @  8.95    c.     895. 
BORROWING     j  Haulin£»  etc"     1000°      "     (borrowed),  @  3.225  c.     322.50 
AND  WASTING.  1          "  1000°      "     (wasted),       @  3.225  c.     322.50 

$2435.00 



These  costs  are  practically  balanced,  but  no  allowance  has 
been  made  for  right  of  way.  If  any  considerable  amount  had 
to  be  paid  for  that,  it  would  decide  this  particular  case  in  favor 
of  the  long  haul.  This  shows  that  under  these  conditions  1800 
feet  is  about  the  limit  of  profitable  haul,  the  land  costing  nothing 
extra. 

BLASTING. 

117.  Explosives.  The  effect  of  blasting  is  due  to  the  ex- 
tremely rapid  expansion  of  a  gas  which  is  developed  by  the 
decomposition  of  a  very  small  amount  of  solid  matter.  Blasting 
compounds  may  be  divided  into  two  general  classes,  (a)  slow- 
burning  and  (5)  detonating.  Gunpowder  is  a  type  of  the  slow- 
burning  compounds.  These  are  generally  ignited  by  heat ;  the 
ignition  proceeds  from  grain  to  grain ;  the  heat  and  pressure 
produced  are  comparatively  low.  Nitro-glycerine  is  a  type  of 
the  detonating  compounds.  They  are  exploded  by  a  shock 
which  instantaneously  explodes  the  whole  mass.  The  heat  and 
pressure  developed  are  far  in  excess  of  that  produced  by  the 
explosion  of  powder.  Nitro-glycerine  is  so  easily  exploded 
that  it  is  very  dangerous  to  handle.  It  was  discovered  that  if 
the  nitro-glycerine  was  absorbed  by  a  spongy  material  like  infu- 


§117.  EARTHWORK.  143 


serial  earth,  it  was  much  less  liable  to  explode,  while  its  power 
when  actually  exploded  was  practically  equal  to  tiiat  of  the 
amount  of  pure  nitre-glycerine  contained  in  the  dynamite,  which 
is  the  name  given  to  the  mixture  of  nitro-glycerine  and  infusorial 
earth.  Nitro-glycerine  is  expensive;  many  other  explosive 
chemical  compounds  which  properly  belong  to  the  slow-burning 
class  are  comparatively  cheap.  It  has  been  conclusively  demon- 
strated that  a  mixture  of  nitro-glycerine  and  some  of  the  cheaper 
chemicals  has  a  greater  explosive  force  than  the  sum  of  the 
strengths  of  the  component  parts  when  exploded  separately. 
Whatever  the  reason,  the  fact  seems  established.  The  reason  is 
possibly  that  the  explosion  of  the  nitro-glycerine  is  sufficiently 
powerful  to  produce  a  detonation  of  the  other  chemicals,  which 
is  impossible  to  produce  by  ordinary  means,  and  that  this  explo- 
sion caused  by  detonation  is  more  powerful  than  an  ordinary 
explosion.  The  majority  of  the  explosive  compounds  and 
"powders"  on  the  market  are  of  this  character — a  mixture  of 
20  to  60  per  cent,  of  nitro-glycerine  with  variable  proportions 
of  one  or  more  of  a  great  variety  of  explosive  chemicals. 

The  choice  of  the  explosive  depends  on  the  character  of  the 
rock.  A  hard  brittle  rock  is  most  effectively  blasted  by  a 
detonating  compound.  The  rapidity  with  which  the  full  force 
of  the  explosive  is  developed  has  a  shattering  effect  on  a  brittle 
substance.  On  the  contrary,  some  of  the  softer  tougher  rocks 
and  indurated  clays  are  but  little  affected  by  dynamite.  The 
result  is  but  little  more  than  an  enlargement  of  the  blast-hole. 
Quarrying  must  generally  be  done  with  blasting-powder,  as  the 
quicker  explosives  are  too  shattering.  Although  the  results 
obtained  by  various  experimenters  are  very  variable,  it  may  be 
said  that  pure  nitro-glycerine  is  eight  times  as  powerful  as  black 
powder,  dynamite  (75$  nitro-glycerine)  six  times,  and  gun-cotton 
four  to  six  times  as  powerful.  For  open  work  where  time  is  not 
particularly  valuable,  black  powder  is  by  far  the  cheapest,  but 
in  tunnel-headings,  whose  progress  determines  the  progress  of 
the  whole  work,  dynamite  is  so  much  more  effective  and  so- 
expedites  the  work  that  its  use  becomes  economical. 


144  RAILROAD  CONSTRUCTION.  §  118. 

118.  Drilling.  Although  many  very  complicated  forms  of 
drill-bars  have  been  devised,  the  best  form  (with  slight  modifi- 
cations to  suit  circumstances)  is  as  shown  in  Fig.  64,  (a)  and  (J). 


FIG.  04. 

The  width  should  flare  at  the  bottom  (a)  about  15  to  30$.  For 
hard  rock  the  curve  of  the  edge  should  be  somewhat  natter  and 
for  soft  rock  somewhat  more  curved  than  shown,  Fig.  64,  (a). 
Sometimes  the  angle  of  the  two  faces  is  varied  from  that  given, 
Fig.  64,  (5),  and  occasionally  the  edge  is  purposely  blunted  so 
as  to  give  a  crushing  rather  than  a  cutting  effect.  The  drills 
will  require  sharpening  for  each  6  to  18  inches  depth  of  hole, 
and  will  require  a  new  edge  to  be  worked  every  2  to  4  days. 
For  drilling  vertical  holes  the  churn-drill  is  the  most  econom- 
ical. The  drill-bar  is  of  iron,  about  6  to  8  feet  long,  1 J"  in 
diameter,  weighs  about  25  to  30  Ibs.,  and  is  shod  with  a  piece 
of  steel  welded  on.  The  bar  is  lifted  a  few  inches  between  each 
blow,  turned  partially  around,  and  allowed  to  fall,  the  impact 
doing  the  work.  From  5  to  15  feet  of  holes,  depending  on  the 
character  of  the  rock,  is  a  fair  day's  work — 10  hours.  In  very 
soft  rocks  even  more  than  this  may  be  done.  This  method  is 
inapplicable  for  inclined  holes  or  even  for  vertical  holes  in  con- 
fined places,  such  as  tunnel-headings.  For  such  places  the  only 
practical  hand  method  is  to  use  hammers.  This  may  be  done 
by  light  drills  and  light  hammers  (one-man  work),  or  by  heavier 
drills  held  by  one  man  and  struck  by  one  or  two  men  with 
heavy  hammers.  The  conclusion  of  an  exhaustive  investigation 
as  to  the  relative  economy  of  light  or  heavy  hammers  is  that  the 
light-hammer  method  is  more  economical  for  the  softer  rocks, 
the  heavy-hammer  method  is  more  economical  for  the  harder 


§119. 


EARTHWORK. 


145 


rocks,  but  that  the  light-hammer  method  is  always  more  ex- 
peditious and  hence  to  be  preferred  when  time  is  important. 

The  subject  of  machine  rock-drills  is  too  vast  to  be  treated 
here.  The  method  is  only  practicable  when  the  amount  of 
work  to  be  done  is  large,  and  especially  when  time  is  valuable. 
The  machines  are  generally  operated  by  compressed  air  for  tun- 
nel-work, thus  doing  the  additional  service  of  supplying  fresh 
air  to  the  tunnel-headings  where  it  is  most  needed.  The  cost 
per  foot  of  hole  drilled  is  quite  variable,  but  is  usually  some- 
what less  than  that  of  hand-drilling — sometimes  but  a  small 
fraction  of  it. 

119.  Position  and  direction  of  drill-holes.  As  the  cost  of 
drilling  holes  is  the  largest  single  item  in  the  total  cost  of  blast- 
ing, it  is  necessary  that  skill  and  judgment  should  be  used  in  so 
locating  the  holes  that  the  blasts  will  be  most  effective.  The 
greatest  effect  of  a  blast  will  evidently  be  in  the  direction  of  the 
"line  of  least  resistance."  In  a  strictly  homogeneous  material 
this  will  be  the  shortest  line  from  the  center  of  the  explosive  to 
the  surface.  The  variations  in  homogeneity  on  account  of 
laminations  and  seams  require  that  each  case  shall  be  judged 
according  to  experience.  In  open- pit  blasting  it  is  generally 
easy  to  obtain  two  and  sometimes  three  exposed  faces  to  the 
rock,  making  it  a  simple  matter  to  drill  holes  so  that  a  blast  will 
do  effective  work.  "VYhen  a  solid  face  of  rock  must  be  broken 
into,  as  in  a  tunnel-heading,  the 
work  is  necessarily  ineffectual  and 
expensive.  A  conical  or  wedge- 
shaped  mass  will  first  be  blown  out 
by  simultaneous  blasts  in  the  holes 
marked  1,  Fig.  65;  blasts  in  the 
holes  marked  2  and  3  will  then  com- 
plete the  cross-section  of  the  head- 

DRILL  HOLES  IN  TUNNEL  HEADING 

ing.     A  great  saving  in  cost  may  FlG  65< 

often  be  secured  by  skilfully  taking 

advantage  of  seams,  breaks,  and  irregularities.    When  the  work 

is  economically  done  there  is  but  little  noise  or  throwing  of  rock, 


146  RAILROAD  CONSTRUCTION.  §  120; 

a  covering  of  old  timbers  and  branches  of  trees  generally  sufficing 
to  confine  the  smaller  pieces  which  would  otherwise  fly  up. 

120.  Amount  of  explosive.  The  amount  of  explosive  required 
varies  as  the  cube  of  the  line  of  least  resistance.  The  best 
results  are  obtained  when  the  line  of  least  resistance  is  f  of  the 
depth  of  the  hole ;  also  when  the  powder  fills  about  -J  of  the 
hole.  For  average  rock  the  amount  of  powder  required  is  as 
follows : 


Line  of  least  resistance  

2  ft 

4  ft 

6  ft 

8  ft 

Weight  of  powder     , 

i  lb 

2  Ibs 

6f  Ibs 

16  Ibs 

Strict  compliance  with  all  of  the  above  conditions  would  re- 
quire that  the  diameter  of  the  hole  should  vary  for  every  case. 
While  this  is  impracticable,  there  should  evidently  be  some 
variation  in  the  size  of  the  hole,  depending  on  the  work  to  be 
done.  For  example,  a  V  hole,  drilled  2'  8"  deep,  with  its 
line  of  least  resistance  2',  and  loaded  with  J  lb.  of  powder,, 
would  be  filled  to  a  depth  of  9J",  which  is  nearly  -J-  of  the 
depth.  A  3"  hole,  drilled  8'  deep,  with  its  line  of  least  resist- 
ance 6',  and  loaded  with  6f  Ibs.  of  powder,  would  be  filled  to 
a  depth  of  over  28",  which  is  also  nearly  J-  of  the  depth.  One 
pound  of  blasting-powder  will  occupy  about  28  cubic  inches. 
Quarrying  necessitates  the  use  of  numerous  and  sometimes 
repeated  light  charges  of  powder,  as  a  heavy  blast  or  a  powerful 
explosive  like  dynamite  is  apt  to  shatter  the  rock.  This 
requires  more  powder  to  the  cubic  yard  than  blasting  for  mere 
excavation,  which  may  usually  be  done  by  the  use  of  J  to  -J-  lb. 
of  powder  per  cubic  yard  of  easy  open  blasting.  On  account 
of  the  great  resistance  offered  by  rock  when  blasted  in  headings 
in  tunnels,  the  powder  used  per  cubic  yard  will  run  up  to  2,  4r 
and  even  6  Ibs.  per  cubic  yard.  As  before  stated,  nitro- 
glycerine is  about  eight  times  (and  dynamite  about  six  times)  as 
powerful  as  the  same  weight  of  powder. 

121.  Tamping.  Blasting-powder  and  the  slow-burning  ex- 
plosives require  thorough  tamping.  Clay  is  probably  the  best, 


§  123  EARTHWORK.  147 

but  sand  and  fine  powdered  rock  are  also  used.  Wooden  plugs, 
inverted  expansive  cones,  etc.,  are  periodically  reinvented  by 
enthusiastic  inventors,  only  to  be  discarded  for  the  simpler 
methods.  Owing  to  the  extreme  rapidity  of  the  development 
of  the  force  of  a  nitro-glycerme  or  dynamite  explosion,  tamping 
is  not  so  essential  with  these  explosives,  although  it  unquestion- 
ably adds  to  their  effectiveness.  Blasting  under  water  has  been 
effectively  accomplished  by  merely  pouring  nitro-glycerine  into 
the  drilled  holes  through  a  tube  and  then  exploding  the  charge 
without  any  tamping  except  that  furnished  by  the  superincum- 
bent water.  It  has  been  found  that  air-spaces  about  a  charge 
make  a  material  reduction  in  the  effectiveness  of  the  explosion. 
It  is  therefore  necessary  to  carefully  ram  the  explosive  into  a 
solid  mass.  Of  course  the  liquid  nitro-glycerine  needs  no  ram- 
ming, but  dynamite  should  be  rammed  with  a  wooden  rammer. 
Iron  should  be  carefully  avoided  in  ramming  gunpowder.  A 
copper  bar  is  generally  used. 

122.  Exploding  the  charge.     Black  powder  is  generally  ex- 
ploded by  means  of  a  fuse  which  is  essentially  a  cord  in  which 
there  is  a  thin  vein  of  gunpowder,  the  cord  being  protected  by 
tar,  extra  linings  of  hemp,  cotton,  or  even  gutta-percha.     The 
fuse  is  inserted  into  the  middle  of  the  charge,  and  the  tamping 
carefully  packed  around  it  so  that  it  will  not  be  injured.     To 
produce  the  detonation  required  to  explode  nitro-glycerine  and 
dynamite,    there  must  be  an  initial  explosion  of  some  easily 
ignited  explosive.     This  is  generally  accomplished  by  means  of 
caps    containing   fulminating-powder    which    are    exploded   by 
electricity.     The  electricity  (in  one  class  of  caps)  heats  a  very 
iine  platinum   wire  to  redness,   thereby  igniting  the  sensitive 
powder,  or  (in  another  class)  a  spark  is  caused  to  jump  through 
the  powder  between  the  ends  of  two  wires  suitably  separated. 
Dynamite  can  also  be  exploded  by  using  a  small  cartridge  of 
gunpowder  which  is  itself  exploded  by  an  ordinary  fuse. 

123.  Cost.     Trautwine  estimates   the    cost   of   blasting  (for 
mere  excavation)  as  averaging  45  cents  per  cubic  yard,   falling 
as  low  as  30  cents  for  easy  but  brittle  rock,  and  running  up  to 


148  RAILROAD   CONSTRUCTION.  §  124. 

60  cents  and  even  $1  when  the  cutting  is  shallow,  the  rock 
especially  tough,  and  the  strata  unf avorably  placed.  Soft  tough 
rock  frequently  requires  more  powder  than  harder  brittle 
rock. 

124.  Classification  of  excavated  material.  The  classification  of 
excavated  material  is  a  fruitful  source  of  dispute  between  con- 
tractors  and  railroad  companies,  owing  mainly  to  the  fact  that 
the  variation  between  the  softest  earth  and  the  hardest  rock  is 
so  gradual  that  it  is  very  difficult  to  describe  distinctions  between 
different  classifications  which  are  unmistakable  and  indisputable. 
The  classification  frequently  used  is  (a)  earth,  (Z>)  loose  rock,  and 
(c)  solid  rock.  As  blasting  is  frequently  used  to  loosen  ' '  loose 
rock  ' '  and  even  ' '  earth  ' '  (if  it  is  frozen),  the  fact  that  blasting 
is  employed  cannot  be  used  as  a  criterion,  especially  as  this 
would  (if  allowed)  lead  to  unnecessary  blasting  for  the  sake  of 
classifying  material  as  rock. 

Earth.  This  includes  clay,  sand,  gravel,  loam,  decomposed 
rock  and  slate,  boulders  or  loose  stones  not  greater  than  1  cubic 
foot  (3  cubic  feet,  P.  R.  R.),  and  sometimes  even  "hard-pan." 
In  general  it  will  signify  material  which  can  be  loosened  by  a 
plough  with  two  horses,  or  with  which  one  picker  can  keep  one 
shoveller  busy. 

Loose  rock.  This  includes  boulders  and  loose  stones  of  more 
than  one  cubic  foot  and  less  than  one  cubic  yard ;  stratified  rock, 
not  more  than  six  inches  thick,  separated  by  a  stratum  of  clay ; 
also  all  material  (not  classified  as  earth)  which  may  be  loosened 
by  pick  or  bar  and  which  "can  be  quarried  without  blasting, 
although  blasting  may  occasionally  be  resorted  to. ' ' 

Solid  rock  includes  all  rock  found  in  masses  of  over  one  cubic 
yard  which  cannot  be  removed  except  by  blasting. 

It  is  generally  specified  that  the  engineer  of  the  railroad 
company  shall  be  the  judge  of  the  classification  of  the  material, 
but  frequently  an  appeal  is  taken  from  his  decisions  to  the  courts. 

125.  Specifications  for  earthwork.  The  following  specifica- 
tions, issued  by  the  Norfolk  and  "Western  R.  R.,  represent  the 
average  requirements.  It  should  be  remembered  that  very  strict 


§  125.  EARTHWORK.  149 

specificatioDS  invariably  increase  the  cost  of  the  work,  and  fre- 
quently add  to  the  cost  more  than  is  gained  by  improved  quality 
of  work. 

1.  The  grading  will  be  estimated  and  paid  for  by  the  cubic 
yard,  and  will  include  clearing  and  grubbing,  and  all  open  ex- 
cavations, channels,  and  embankments  required  for  the  forma- 
tion of  the  roadbed,  and  for  turnouts  and  sidings ;   cutting  all 
ditches  or  drains  about  or  contiguous  to  the  road ;   digging  the 
foundation-pits  of  all  culverts,  bridges,  or  walls ;  reconstructing 
turnpikes  or  common  roads  in  cases  where  they  are  destroyed  or 
interfered  with ;  changing  the  course  or  channel  of  streams ;  and 
all  other  excavations  or  embankments  connected  with  or  incident 
to  the  construction  of  said  Railroad. 

2.  All  grading,  except  where  otherwise  specified,  whether 
for  cuts  or  fills,  will  be  measured  in  the  excavations  and'  will  be 
classified  under  the   following  heads,  viz.  :   Solid  Rock,  Loose 
Rock,  Hard-pan,  and  Earth. 

SOLID  ROCK  shall  include  all  rock  occurring  in  masses  which, 
in  the  judgment  of  the  said  Engineer  Maintenance  of  Way,  may 
be  best  removed  by  blasting. 

LOOSE  ROCK  shall  include  all  kinds  of  shale,  soapstone,  and 
other  rock  which,  in  the  judgment  of  the  said  Engineer  Main- 
tenance of  Way,  can  be  removed  by  pick  and  bar,  and  is  soft 
and  loose  enough  to  be  removed  without  blasting,  although 
blasting  may  be  occasionally  resorted  to ;  also,  detached  stone  of 
less  than  one  (1)  cubic  yard  and  more  than  one  (1)  cubic  foot. 

HARD-PAN  shall  consist  of  tough  indurated  clay  or  cemented 
gravel,  which  requires  blasting  or  other  equally  expensive 
means  for  its  removal,  or  which  cannot  be  ploughed  with  less 
than  four  horses  and  a  railroad  plough,  or  which  requires  two 
pickers  to  a  shoveller,  the  said  Engineer  Maintenance  of  Way 
to  be  the  judge  of  these  conditions. 

EARTH  shall  include  all  material  of  an  earthy  nature,  of 
whatever  name  or  character,  not  unquestionably  loose  rock  or 
hard-pan  as  above  defined. 

POWDER.     The  use  of  powder  in  cuts  will  not  be  considered 


150  RAILROAD  CONSTRUCTION.  §  125 

as  a  reason  for  any  other  classification  than  earth,  unless  the 
material  in  the  cut  is  clearly  other  than  earth  under  the  above 
specifications. 

3.  Earth,  gravel,  and  other  materials  taken  from  the  exca- 
vations, except  when  otherwise  directed  by  the  said  Engineer 
Maintenance  of  Way  or  his  assistant,  shall  be  deposited  in  the 
adjacent  embankment;    the  cost   of  removing   and  depositing 
which,  when  the  distance  necessary  to  be  hauled  is  not  more 
than  sixteen  hundred  (1600)  feet,  shall  be  included  in  the  price 
paid  for  the  excavation. 

4.  EXTRA  HAUL  will  be  estimated  and  paid  for  as  follows : 
whenever  material  from   excavations   is   necessarily   hauled    a 
greater  distance  than  sixteen  hundred  (1600)  feet,  there  shall  be 
paid  in  addition  to  the  price  of  excavation  the  price  of  extra 
haul  per  100  feet,  or  part  thereof,  after  the  first  1600  feet;   the 
necessary  haul  to  be  determined  in  each  case  by  the  said  Engi- 
neer Maintenance  of  Way  or  his  assistant,  from  the  profile  and 
cross- sections,  and  the  estimates  to  be  in  accordance  therewith. 

5.  All  embankments  shall  be  made  in  layers  of  such  thick- 
ness and  carried  on  in  such  manner  as  the  said  Engineer  Mainte- 
nance of  Way  or  his  assistant  may  prescribe,   the  stone  and 
heavy  materials  being  placed  in  slopes  and  top.     And  in  com- 
pleting the  fills  to  the  proper  grade  such  additional  heights  and 
fulness  of  slope  shall  be  given  them,  to  provide  for  their  settle- 
ment, as  the  said  Engineer  Maintenance  of  Way,  or  his  assistant, 
may  direct.     Embankments  about  masonry  shall    be  built   at 
such  times  and  in  such  manner  and  of  such  materials  as  the  said 
Engineer  Maintenance  of  Way  or  life  assistant  may  direct. 

6.  In  procuring  materials  for  embankments  from  without 
the  line  of  the  road,  and  in  wasting  materials  from  cuttings,  the 
place  and  manner  of  doing  it  shall  in  each  case  be  indicated  by 
the  Engineer  Maintenance  of  Way  or  his  assistant;    and  care 
must  be  taken  to  injure  or  disfigure  the  land  as  little  as  possible. 
Borrow-pits  and  spoil-banks  must  be  left  by  the  Contractor  in 
regular  and  sightly  shape. 

7.  The  lands  of  the  said  Eailroad  Company  shall  be  cleared 


§  125.  EARTHWORK.  151 

to  the  extent  required  by  the  said  Engineer  Maintenance  of 
Way,  or  his  assistant,  of  all  trees,  brushes,  logs,  and  other 
perishable  materials,  which  shall  be  destroyed  by  burning  or 
deposited  in  heaps  as  the  said  Engineer  Maintenance  of  Way, 
or  his  assistant,  may  direct.  Large  trees  must  be  cut  not  more 
than  two  and  one-half  (2£)  feet  from  the  ground,  and  under 
embankments  less  than  four  (4)  feet  high  they  shall  be  cut  close 
to  the  ground.  All  small  trees  and  bushes  shall  be  cut  close  to 
the  ground. 

8.  Clearing  shall  be  estimated  and  paid  for  by  the  acre  or 
fraction  of  an  acre. 

9.  All  stumps,  roots,  logs,  and  other  obstructions  shall  be 
grubbed  out,  and  removed  from  all  places  where  embankments 
occur  less  than  two  (2)  feet  in  height;   also,  from  all  places 
where  excavations  occur  and  from  such  other  places  as  the  said 
Engineer  Maintenance  of  Way  or  his  assistant  may  direct. 

10.  Grubbing  shall  be  estimated  and  paid  for  by  the  acre  or 
fraction  of  an  acre. 

11.  Contractors,  when  directed  by  the  said  Engineer  Main- 
tenance of  Way   or  his  assistant  in  charge  of  the  work,  will 
deposit  on  the  side  of  the  road,  or  at  such  convenient  points  as 
may  be  designated,  any  stone,  rock,  or  other  materials  that  they 
may  excavate;    and  all   materials  excavated  and  deposited   as 
above,  together  with  all  timber  removed  from'  the  line  of  the 
road,  will  be  considered  the  property  of  the  Railroad  Company, 
and  the  Contractors  upon  the  respective  sections  will  be  respon- 
sible for  its  safe-keeping  until  removed  by  said  Railroad  Com- 
pany, or  until  their  work  is  finished. 

12.  Contractors  will   be    accountable  for  the  maintenance 
of  safe  and  convenient  places  wherever  public  or  private  roads 
are  in  any  way  interfered  with  by  them  during  the  progress  of 
the  work.      They  will  also  be  responsible  for  fences  thrown 
down,  and  for  gates  and  bars  left  open,  and  for  all  damages 
occasioned  thereby. 

13.  Temporary  bridges  and  trestles,  erected  to  facilitate  the 
progress  of  the  work,  in  case  of  delays  at  masonry  structures 


152  BAILROAD    CONSTRUCTION.  §  125, 

from  any  cause,  or  for  other  reasons,  will  be  at  the  expense  of 
the  Contractor. 

14.  The  line  of  road  or  the  gradients  may  be  changed  in 
any  manner,  and  at  any  time,  if  the  said  Engineer  Maintenance 
of  Way  or  his  assistant  shall  consider  such  a  change  necessary 
or  expedient ;  but  no  claim  for  an  increase  in  prices  of  excava- 
tion or  embankment  on  the  part  of  the  Contractor  will  be  allowed 
or  considered  unless  made  in  writing  before  the  work  on  that 
part  of  the  section  where  the  alteration  has  been  made  shall  have 
been  commenced.      The  said  Engineer  Maintenance  of  Way  or 
his  assistant  may  also,  on  the  conditions  last  recited,  increase  or 
diminish  the   length  of  any  section  for  the  purpose  of  more 
nearly  equalizing  or  balancing  the  excavations  and  embankments, 
or  for  any  other  reason. 

15.  The  roadbed  will  be  graded  as  directed  by  the  said  En- 
gineer Maintenance  of  Way  or  his  assistant,  and  in  conformity 
with  such  breadths,  depths,  and  slopes  of  cutting  and  filling  as 
he  may  prescribe  from  time  to  time,  and  no  part  of  the  work 
will  be  finally  accepted  until  it  is  properly  completed  and  dressed 
off  at  the  required  grade. 


CHAPTER  IV. 

TRESTLES. 

126.  Extent  of  use.  Trestles  constitute  from  1  to  3#  of  the 
length  of  the  average  railroad.  It  was  estimated  in  1889  that 
there  was  then  about  2400  miles  of  single-track  railway  trestle 
in  the  United  States,  divided  among  150,000  structures  and 
estimated  to  cost  about  $75,000,000.  The  annual  charge  for 
maintenance,  estimated  at  -J  of  the  cost,  therefore  amounted  to 
about  $9,500,000  and  necessitated  the  annual  use  of  perhaps 
300,000,000  ft.  B.M.  of  timber.  The  corresponding  figures  at 
the  present  time  must  be  somewhat  in  excess  of  this.  The 
magnitude  of  this  use,  which  is  causing  the  rapid  disappearance 
of  forests,  has  resulted  in  endeavors  to  limit  the  use  of  timber 
for  this  purpose.  Trestles  may  be  considered  as  justifiable  under 
the  following  conditions : 

a.  Permanent  trestles. 

1.  Those  of  extreme  height — then  called  viaducts  and  fre- 
quently constructed  of  iron  or  steel,  as  the  Kinzua  viaduct,  302 
ft.  high. 

2.  Those  across  waterways — e.g.,  that  across  Lake  Pontchar- 
train,  near  !N"ew  Orleans,  22  miles  long. 

3.  Those  across  swamps  of  soft  deep  mud,  or  across  a  river- 
bottom,  liable  to  occasional  overflow. 

b.  Temporary  trestles. 

1.  To  open  the  road  for  traffic  as  quickly  as  possible — often 
a  reason  of  great  financial  importance. 

2.  To  quickly  replace  a  more  elaborate  structure,  destroyed 

153 


154  RAILROAD  CONSTRUCTION.  §127. 

by  accident,  on  a  road  already  in  operation,  so  that  the  inter- 
ruption to  traffic  shall  be  a  minimum.  • 

3.  To  form  an  earth  embankment  with  earth  brought  from 
a  distant  point  by  the  train-load,  when  such  a  measure  would 
cost  less  than  to  borrow  earth  in  the  immediate  neighborhood. 

4.  To  bridge  an  opening  temporarily  and  thus  allow  time  to 
learn  the  regimen  of  a  stream  in  order  to  better  proportion  the 
size  of  the  waterway  and  also  to  facilitate  bringing  suitable  stone 
for  masonry  from  a  distance.      In  a  new  country  there  is  always 
the  double  danger  of  either  building  a  culvert  too  small,  requir- 
ing expensive  reconstruction,  perhaps  after  a  disastrous  washout, 
or  else  wasting  money  by  constructing  the  culvert  unnecessarily 
large.     Much  masonry  has  been  built  of  a  very  poor  quality  of 
stone  because  it  could  be  conveniently  obtained  and  because  good 
stone  was  unobtainable  except  at  a  prohibitive  cost  for  transpor- 
tation.    Opening  the  road  for  traffic  by  the  use  of  temporary 
trestles  obviates  both  of  these  difficulties. 

127.  Trestles  vs.  embankments.  Low  embankments  are  very 
much  cheaper  than  low  trestles  both  in  first  cost  and  mainte- 
nance. Yery  high  embankments  are  very  expensive  to  construct, 
but  cost  comparatively  little  to  maintain.  A  trestle  of  equal 
height  may  cost  much  less  to  construct,  but  will  be  expensive  to 
maintain — perhaps  -J  of  its  cost  per  year.  To  determine  the 
height .  beyond  which.it  will  be  cheaper  to  maintain  a  trestle 
rather  than  build  an  embankment,  it  will  be  necessary  to  allow 
for  the  cost  of  maintenance.  The  height  will  also  depend  on 
the  relative  cost  of  timber,  labor,  and  earthwork.  At  the  pres- 
ent average  values,  it  will  be  found  that  for  less  heights  than 
25  feet  the  first  cost  of  an  embankment  will  generally  be  less 
than  that  of  a  trestle ;  this  implies  that  a  permanent  trestle 
should  never  be  constructed  with  a  height  less  than  25  feet 
except  for  the  reasons  given  in  §  126.  The  height  at  which  a 
permanent  trestle  is  certainly  cheaper  than  earthwork  is  more 
uncertain.  A  high  grade  line  joining  two  hills  will  invariably 
imply  at  least  a  culvert  if  an  embankment  is  used.  If  the 
culvert  is  built  of  masonry,  the  cost  of  the  embankment  will  be 


§  129.  TRESTLES.  155 

so  increased  that  the  height  at  which  a  trestle  becomes  economi- 
cal will  be  materially  reduced.  The  cost  of  an  embankment 
increases  much  more  rapidly  than  the  height — with  very  high 
embankments  more  nearly  as  the  square  of  the  height — while 
the  cost  of  trestles  does  not  increase  as  rapidly  as  the  height. 
Although  local  circumstances  may  modify  the  application  of  any 
set  rules,  it  is  probably  seldom  that  it  will  be  cheaper  to  build 
an  embankment  40  or  50  feet  high  than  to  permanently  maintain 
a  wooden  trestle  of  that  height.  A  steel  viaduct  would  proba- 
bly be  the  best  solution  of  such  a  case.  These  are  frequently 
used  for  permanent  structures,  especially  when  very  high.  The 
cost  of  maintenance  is  much  less  than  that  of  wood,  which 
makes  the  use  of  iron  or  steel  preferable  for  permanent  trestles 
unless  wood  is  abnormally  cheap.  Xeither  the  cost  nor  the  con- 
struction of  iron  or  steel  trestles  will  be  considered  in  this  chapter. 

128.  Two  principal  types.     There  are  two  principal  types  of 
wooden  trestles — pile  trestles  and  framed  trestles.     The  great 
objection  to  pile  trestles  is  the  rapid  rotting  of  the  portion  of 
the  pile  which  is  underground,  and  the  difficulty  of  renewal. 
The  maximum  height  of  pile  trestles  is  about  30  feet,  and  even  this 
height  is  seldom  reached.     Framed  trestles  have  been  constructed 
to  a  height  of  considerably  over  100  feet.     They  are  frequently 
built  in  such  a  manner  that  any  injured  piece  may  be  readily 
taken  out  and  renewed  without  interfering  with  traffic.     Trestles 
consist  of   two  parts — the   supports  called   "bents,"   and   the 
stringers  and  floor  system.     As  the  stringers  and  floor  system 
are  the  same  for  both  pile  and  framed  trestles,  the  ' '  bents ' '  are 
all  that  need  be  considered  separately. 

PILE    TRESTLES. 

129.  Pile  bents.     A  pile  bent  consists  generally  of  four  piles 
driven  into  the  ground  deep  enough  to  afford  not  only  sufficient 
vertical  resistance  but  also  lateral  resistance.      On  top  of  these 
piles  is  placed  a  horizontal  "cap."     The  caps  are  fastened  to 
the  tops  of  the  piles  by  methods  illustrated  in  Fig.  66.     The 


156 


RAILROAD  CONSTRUCTION. 


§129. 


method  of  fastening  shown  in  each  case  should  not  be  considered 
as  applicable  only  to  the  particular  type  of  pile  bent  used  to  illus- 
trate it.  Fig.  66  (a  and  d)  illustrates  a  mortise- joint  with  a  hard- 


FIG.  06. 

wood  pin  about  1 J"  in  diameter.  The  hole  for  the  pin  should 
be  bored  separately  through  the  cap  and  the  mortise,  and 
the  hole  through  the  cap  should  be  at  a  slightly  higher 
level  than  that  through  the  mortise,  so  that  the  cap  will  be 
drawn  down  tight  when  the  pin  is  driven.  Occasionally  an 
iron  dowel  (an  iron  pin  about  1  j-"  in  diameter  and  about  6" 
long)  is  inserted  partly  in  the  cap  and  partly  in  the  pile.  The 
use  of  drift-bolts,  shown  in  Fig.  66  (7>),  is  cheaper  in  first  cost,  but 
renders  repairs  and  renewals  very  troublesome  and  expensive. 
u  Split  caps,"  shown  in  Fig.  66  (V),  are  formed  by  bolting  two 
half-size  strips  on  each  side  of  a  tenon  on  top  of  the  pile. 
.Repairs  are  very  easily  and  cheaply  made  without  interference 
with  the  traffic  and  without  injuring  other  pieces  of  the  bent. 
The  smaller  pieces  are  more  easily  obtainable  in  a  sound  con- 
dition ;  the  decay  of  one  does  not  affect  the  other,  and  the  first 
cost  is  but  little  if  any  greater  than  the  method  of  using  a  single 
piece.  For  further  discussion,  see  §  186. 

For  very  light  traffic  and  for  a  height  of  about  5  feet  three 
vertical  piles  will  suffice,  as  shown  in  Fig.  66  (a).  Up  to  a  height 
of  8  or  10  feet  four  piles  may  be  used  without  sway- bracing,  as 
in  Fig.  66  (b),  if  the  piles  have  a  good  bearing.  For  heights 
greater  than  10  feet  sway-bracing  is  generally  necessary.  The 
outside  piles  are  frequently  driven  with  a  batter  varying  from 
1 :  12  to  1 :  4. 


§  130.  TRESTLES.  157 

Piles  are  made,  if  possible,  from  timber  obtained  in  the 
vicinity  of  the  work.  Durability  is  the  great  requisite  rather 
than  strength,  for  almost  any  timber  is  strong  enough  (except 
as  noted  below)  and  will  be  suitable  if  it  will  resist  rapid  decay. 
The  following  list  is  quoted  as  being  in  the  order  of  preference 
on  account  of  durability : 


1.  Red  cedar 

2.  Red  cypress 

3.  Pitch-pine 

4.  Yellow  pine 


5.  White  pine 

6.  Redwood 

7.  Elm 

8.  Spruce 


9.  White  oak 

10.  Post-oak 

11.  Red  oak 


12.  Black  oak 

13.  Hemlock 

14.  Tamarac 


Red- cedar  piles  are  said  to  have  an  average  life  of  27  years 
with  a  possible  maximum  of  50  years,  but  the  timber  is  rather 
weak,  and  if  exposed  in  a  river  to  flowing  ice  or  driftwood  is 
apt  to  be  injured.  Under  these  circumstances  oak  is  prefer- 
able, although  its  life  may  be  only  13  to  18  years. 

130.  Methods  of  driving  piles.  The  following  are  the  prin- 
cipal methods  of  driving  piles : 

a.  A  hammer  weighing  2000  to  3000  Ibs.  or  more,  sliding 
in  guides,  is  drawn  up  by  horse-power  or  a  portable  engine,  and 
allowed  to  fd&  freely. 

b.  The  same  as  above  except  that  the  hammer  does  not  fall 
freely,  but  drags  the  rope  and  revolving  drum  as  it  falls  and  is 
thus  quite  materially  retarded.     The  mechanism  is  a  little  more 
simple,  but  is  less  effective,  and  is  sometimes  made  deliberately 
deceptive  by  a  contractor  by  retarding  the  blow,  in  order  to 
apparently   indicate   the   requisite    resistance    on   the   part   of 
the  pile. 

The  above  methods  have  the  advantage  that  the  mechanism 
is  cheap  and  can  be  transported  into  a  new  country  with  com- 
parative ease,  but  the  work  done  is  somewhat  ineffective  and 
costly  compared  with  some  of  the  more  elaborate  methods 
given  below. 

c.  Gunpowder  pile-drivers,  which  automatically  explode  a 
cartridge  every  time  the  hammer  falls.     The  explosion  not  only 
forces  the  pile  down,  but  throws  up  the  hammer  for  the  next 
blow.    For  a  given  height  of  fall  the  effect  is  therefore  doubled. 
It  has  been  shown  by  experience,  however,  that  when  it  is  at- 


158  RAILROAD  CONSTRUCTION.  §  130. 

tempted  to  use  such  a  pile-driver  rapidly  tlie  mechanism  be- 
comes so  heated  that  the  cartridges  explode  prematurely,  and  the 
method  has  therefore  been  abandoned. 

d.  Steam  pile-drivers,  in  which  the  hammer  is  operated 
directly  by  steam.  The  hammer  falls  freely  a  height  of  about 
40  inches  and  is  raised  again  by  steam.  The  effectiveness  is 
largely  due  to  the  rapidity  of  the  blows,  which  does  not  allow 
time  between  the  blows  for  the  ground  to  settle  around  the  pile 
and  increase  the  resistance,  which  does  happen  when  the  blows 
are  infrequent.  "The  hammer-cylinder  weighs  5500  Ibs.,  and 
with  60  to  75  Ibs.  of  steam  gives  75  to  80  blows  per  minute. 
With  41  blows  a  large  unpointed  pile  was  driven  35  feet  into  a 
hard  clay  bottom  in  half  a  minute."  Such  a  driver  would  cost 
about  $800. 

The  above  four  methods  are  those  usual  for  dry  earth. 
In  very  soft  wet  or  sandy  soils,  where  an  unlimited  supply  of 
water  is  available,  the  water-jet  is  sometimes  employed.  A  pipe 
is  fastened  along  the  side  of  the  pile  and  extends  to  the  pile- 
point.  If  water  is  forced  through  the  pipe,  it  loosens  the  sand 
around  the  point  and,  rising  along  the  sides,  decreases  the  side 
resistance  so  that  the  pile  sinks  by  its  own  weight,  aided  perhaps 
by  extra  weights  loaded  on.  This  loading  may  be  accomplished 
by  connecting  the  top  of  the  pile  and  the  pile-driver  by  a  block 
and  tackle  so  that  a  portion  of  the  weight  of  the  pile-driver  is 
continually  thrown  on  the  pile. 

Excessive  driving  frequently  fractures  the  pile  below  the 
surface    and  thereby   greatly   weakens  its  bearing  power.      To 
prevent  excessive   "brooming"  of  the  top  of  the 
pile,  owing  to  the  action  of  the  hammer,  the  top 
should  be  protected  by  an  iron  ring  fitted  to  the  top 
of  the  pile.     The  "brooming"  not  only  renders  the 
driving   ineffective    and   hence    uneconomical,    but 
vitiates  the  value  of  any  test  of  the  bearing  power 
of  the  pile  by  noting  the  sinking  due  to  a  given 
FIG.  67.       weight  falling  a  given  distance.     If  the  pile  is  so 
soft  that  brooming  is  unavoidable,  the  top  should  be  adzed  off 


§  131.  TRESTLES.  159 

frequently,  and  especially  should  it  be  done  just  before  the  final 
blows  which  are  to  test  its  bearing-power. 

In  a  new  country  judgment  and  experience  will  be  required 
to  decide  intelligently  whether  to  employ  a  simple  drop-hammer 
machine,  operated  by  horse-power  and  easily  transported  but 
uneconomical  in  operation,  or  a  more  complicated  machine 
working  cheaply  and  effectively  after  being  transported  at 
greater  expense. 

131.  Pile-driving  formulae.  If  R  =  the  resistance  of  a  pile, 
and  s  the  set  of  the  pile  during  the  last  blow,  iv  the  weight  of 
the  pile-hammer,  and  h  the  fall  during  the  last  blow,  then  we 

may  state  the  approximate  relation  that  Rs  —  wh,  or  R  =  — . 

s 

This  is  the  basic  principle  of  all  rational  formulae,  but  the 
maximum  weight  which  a  pile  will  sustain  after  it  has  been 
driven  some  time  is  by  no  means  equal  to  the  resistance  of  the 
pile  during  the  last  blow.  There  are  also  many  other  modi- 
fying elements  which  have  been  variously  allowed  for  in  the 
many  proposed  formulae.  The  formulae  range  from  the  extreme 
of  empirical  simplicity  to  very  complicated  attempts  to  allow 
properly  for  all  modifying  causes.  As  the  simplest  rule, 
specifications  sometimes  require  that  the  piles  shall  be  driven 
until  the  pile  will  not  sink  more  than  5  inches  under  five 
consecutive  blows  of  a  2000  lb.,  hammer  falling  25  feet. 
The  ' i  Engineering  Neivs  formula ' '  *  gives  the  safe  load  as 

— ,    in    which    iv  =  weight    of   hammer,    7i  =  fall  in  feet, 

s  =  set  of  pile  in  inches  under  the  last  blow.  This  formula  is 
derived  from  the  above  basic  formula  by  calling  the  safe  load  -J- 
of  the  final  resistance,  and  by  adding  (arbitrarily)  1  to  the  final 
set  (s)  as  a  compensation  for  the  extra  resistance  caused  by  the 
settling  of  earth  around  the  pile  between  each  blow.  This 
formula  is  used  only  for  ordinary  hammer-driving.  When  the 
piles  are  driven  by  a  steam  pile-driver  the  formula  becomes 

*  Engineering  News,  Nov.  17,  1892. 


160 


RAILROAD  CONSTRUCTION. 


§132. 


Zwh 


safe  load  =      ,  Foi   the  "  gunpowder  pile-driver,"  since 

the  explosion  of  the  cartridge  drives  the  pile  in  with  the  same 
force  with  which  it  throws  the  hammer  upward,  the  effect  is 
twice  that  of  the  fall  of  the  hammer,  and  the  formula  becomes 

safe  load  =  -.     In  these  last  two  formulae  the  constant 

in  the  denominator  is  changed  from  s  +  1  to  s  +  0.1.  The 
constant  (1.0  or  0.1)  is  supposed  to  allow,  as  before  stated,  for  the 
effect  of  the  extra  resistance  caused  by  the  earth  settling  around  the 
pile  between  each  blow.  The  more  rapid  the  blows  the  less  the 
opportunity  to  settle  and  the  less  the  proper  value  of  the  constant. 
The  above  formulae  have  been  given  on  account  of  their 
simplicity  and  their  practical  agreement  with  experience.  Many 
other  formulae  have  been  proposed,  the  majority  of  which  are 
more  complicated  and  attempt  to  take  into  account  the  weight  of 
the  pile,  resistance  of  the  guides,  etc.  While  these  elements, 
as  well  as  many  others,  have  their  influence,  their  effect  is  so 
overshadowed  by  the  indeterminable  effect  of  other  elements — as, 
for  example,  the  effect  of  the  settlement  of  earth  around  the  pile 
between  blows — that  it  is  useless  to  attempt  to  employ  anything 
but  a  purely  empirical  formula. 

132.  Pile-points  and  pile-shoes.  Piles  are  generally  sharpened 
to  a  blunt  point.  If  the  pile  is  liable  to  strike  boulders,  sunken 
logs,  or  other  obstructions  which  are  liable  to  turn  the  point,  it 
is  necessary  to  protect  the  point  by  some 
form  of  shoe.  Several  forms  in  cast  iron 
have  been  used,  also  a  wrought-iron  shoe, 
having  four  ' '  straps  ' '  radiating  from  the 
apex,  the  straps  being  nailed  on  to  the  pile, 
as  shown  in  Fig.  68  (b).  The  cast-iron 
form  shown  in  Fig.  68  (a)  has  a  base  cast 
around  a  drift-bolt.  The  recess  on  the  top 
of  the  base  receives  the  bottom  of  the  pile 
and  prevents  a  tendency  to  split  the  bottom 
of  the  pile  or  to  force  the  shoe  off  laterally. 


FIG.  68. 


§  134.  TRESTLES.  161 

133.  Details  of  design.     Ko  theoretical  calculations  of  the 
strength  of  pile  bents  need  be  attempted  on  account  of  the  ex- 
treme complication  of  the  theoretical  strains,  the  uncertainty  as 
to  the  real  strength  of  the  timber  used,  the  variability  of  that 
strength  with  time,  and  the  insignificance  of  the  economy  that 
would  be  possible  even  if  exact  sizes  could  be  computed.     The 
piles  are  generally  required  to  be  not  less  than  10"  or  12"  in 
diameter  at  the  large  end.    The  P.  R.  R.  requires  that  they  shall 
be  "  not  less  than  14  and  7  inches  in  diameter  at  butt  and  small 
end  respectively,  exclusive  of  bark,  which  must  be  removed." 
The  removal  of  the  bark  is  generally  required  in  good  work. 
Soft  durable  woods,  such  as  are  mentioned  in  §  129,  are  best 
for  the  piles,  but  the  caps  are  generally  made  of  oak  or  yellow 
pine.     The  caps  are  generally   14  feet  long  (for  single  track) 
with  a  cross-section  12"  X  12"  or  12"  X  14".      "  Split  caps" 
would  consist  of  two  pieces  6"  X  12".     The  sway-braces,  never 
used  for  less  heights  than  6',  are  made  of  3"  X  12"  timber,  and 
are  spiked  on  with  f  "  spikes  8''  long.     The  floor  system  will  be 
the  same  as  that  described  later  for  framed  trestles. 

134.  Cost  of  pile  trestles.    The  cost,  per  linear  foot,  of  piling 
depends  on  the  method  of  driving,  the  scarcity  of  suitable  tim- 
ber, the  price  of  labor,  the  length  of  the  piles,  and  the  amount 
of  shifting  of  the  pile-driver  required.     The  cost  of  soft-wood 
piles  varies  from  8  to  15  c.  per  lineal  foot,  and  the  cost  of  oak 
piles  varies  from  10  to  30  c.  per  foot  according  to  the  length, 
the  longer  piles  costing  more  per  foot.     The  cost  of  driving  will 
average  about  82.50  per  pile,  or  7.5  to  10  c.  per  lineal  foot. 
Since  the  cost  of  shifting  the  pile-driver  is  quite  an  item  in  the 
total  cost,  the  cost  of  driving  a  long  pile  would  be  less  per  foot 
than  for  a  short  pile,  but  on  the  other  hand  the  cost  of  the  pile 
is   greater   per   foot,  which    tends  to  make  the  total  cost  per 
foot  constant.      Specifications  generally  say  that  the  piling  will 
be  paid  for  per  lineal  foot  of  piling  left  in  the  work.    The  wast- 
age of  the  tops  of  piles  sawed  off  is  always  something,  and  is 
frequently  very  large.     Sometimes  a  small  amount  per  foot  of 
piling  sawed  off  is  allowed  the  contractor  as  compensation  for 


162 


RAILROAD  CONSTRUCTION. 


135. 


liis  loss.  This  reduces  the  contractor's  risk  and  possibly  reduces 
his  bid  by  an  equal  or  greater  amount  than  the  extra  amount 
actually  paid  him. 


FRAMED    TRESTLES. 


135.  Typical  Design.  A  typical  design  for  a  framed  trestle 
bent  is  given  in  Fig.  69.  This  represents,  with  slight  variations 
of  detail,  the  plan  according  to  which  a  large  part  of  the  framed 


FIG.  69. 

trestle  bents  of  the  country  have  been  built — i.e.,  of  those  less 
than  20  or  30  feet  in  height,  not  requiring  multiple-story 
construction. 

136.  Joints,  (a)  The  mortise-and-tenon  joint  is  illustrated  in 
Fig.  69  and  also  in  Fig.  66  (a).  The  tenon  should  be  about 
3"  thick,  8"  wide,  and  5-J"  long.  The  mortise  should  be  cut 
a  little  deeper  than  the  tenon.  "  Drip-holes  "  from  the  mortise 
to  the  outside  will  assist  in  draining  off  water  that 
may  accumulate  in  the  joint  and  thus  prevent  the 
rapid  decay  that  would  otherwise  ensue.  These 
joints  are  very  troublesome  if  a  single  post  decays 
and  requires  renewal.  It  is  generally  required  that 
FIG.  70.  the  mortise  and  tenon  should  be  thoroughly  daubed 
with  paint  before  putting  them  together.  This  will  tend  to 


§  137. 


TRESTLES. 


163 


FIG.  71. 


make  the  joint  water- tight  and  prevent  decay  from  the  ac- 
cumulation and  retention  of  water  in  the  joint. 

(b)  The  plaster  joint.     This  joint  is  made   by   bolting   and 
spiking  a  3"  X  12"  plank  on  both 

sides  of  the  joint.  The  cap  and 
sill  should  be  notched  to  receive 
the  posts.  Repairs  are  greatly 
facilitated  by  the  use  of  these 
joints.  This  method  has  been 
used  by  the  Delaware  and  Hud- 
son Canal  Co.  [R.  R.]. 

(c)  Iron  plates.    An  iron  plate  of  the  form  shown  in  Fig.  72 
(b)  is  bent  and  used  as  shown  in  Fig.  72  (a).    Bolts  passing  through 

the  bolt-holes  shown  secure  the 
plates  to  the  timbers  and  make  a 
strong  joint  which  may  be  readily 
loosened  for  repairs.  By  slight 
c  modifications  in  the  design  the 
method  may  be  used  for  inclined 
posts  and  complicated  joints. 

(d)  Split  caps  and  sills.    These 
FIG  72.  are  described  in   §   129.      Their 

advantages  apply  with  even  greater  force  to  framed  trestles. 

(e)  Dowels  and  drift-bolts.  These  joints  facilitate  cheap  and 
rapid  construction,  but  renewals  and  repairs  are  very  difficult,  it 
being  almost  impossible  to  extract  a  drift-bolt  which  has  been 
driven  its  full  length  without  splitting  open  the  pieces  contain- 
ing it.  Notwithstanding  this  objection  they  are  extensively 
used,  especially  for  temporary  work  which  is  not  expected  to 
be  used  long  enough  to  need  repairs. 

137.  Multiple-story  construction.  Single-story  framed  trestle 
bents  are  used  for  heights  up  to  18  or  20  feet  and  exceptionally 
up  to  30  feet.  For  greater  heights  some  such  construction 
as  is  illustrated  in  a  skeleton  design  in  Fig.  73  is  used.  By 
using  split  sills  between  each  story  and  separate  vertical  and 
batter  posts  in  each  story,  any  piece  may  readily  be  removed  and 


164 


RAILROAD  CONSTRUCTION. 


138. 


renewed  if  necessary.  .The  height  of  these  stories  varies,  in 
different  designs,  from  15  to  25  and 
even  30  feet.  In  some  designs  the 
structure  of  each  story  is  independent 
of  the  stories  above  and  below.  This 
greatly  facilitates  both  the  original  con- 
struction and  subsequent  repairs.  In 
other  designs  the  verticals  and  batter- 
posts  are  made  continuous  through  two 
consecutive  stories.  The  structure  is 
somewhat  stiffer,  but  is  much  more  diffi- 
cult to  repair. 

Since  the  bents  of  any  trestle  are 
usually  of  variable  height  and  those 
heights  are  not  always  an  even  multiple 


FIG.  73. 


of    the    uniform    height    desired    for   the    stories,   it    becomes 
necessary  to  make  the  upper  stories  of  uniform  height  and  let 


FIG.  74. 

the  odd  amount  go  to  the  lowest  story,  as  shown  in  Figs.  73 
and  74. 

138.  Span.  The  shorter  the  span  the  greater  the  number 
of  trestle  bents ;  the  longer  the  span  the  greater  the  required 
strength  of  the  stringers  supporting  the  floor.  Economy  de- 
mands the  adoption  of  a  span  that  shall  make  the  sum  of  these 
requirements  a  minimum.  The  higher  the  trestle  the  greater 
the  cost  of  each  bent,  and  the  greater  the  span  that  would  be 
justifiable.  Nearly  all  trestles  have  bents  of  variable  height, 
but  the  advantage  of  employing  uniform  standard  sizes  is  so 
great  that  many  roads  use  the  same  span  and  sizes  of  timber  not 
only  for  the  panels  of  any  given  trestle,  but  also  for  all  trestles 


139. 


TRESTLES. 


165 


regardless  of  height.  The  spans  generally  used  vary  from  10 
to  16  feet.  The  Norfolk  and  Western  R.  R.  uses  a  span  of 
12'  6"  for  all  single-story  trestles,  and  a  span  of  25'  for 
all  multiple-story  trestles.  The  stringers  are  the  same  in  both 
cases,  but  when  the  span  is  25  feet,  knee- braces  are  run 


FIG.  75. 

from  the  sill  of  the  first  story  below  to  near  the  middle  of  each 
set  of  stringers.  These  knee-braces  are  connected  at  the  top  by 
a  ' '  straining-beam  ' '  on  which  the  stringers  rest,  thus  support- 
ing the  stringer  in  the  center  and  virtually  reducing  the  span 
about  one-half. 

139.  Foundations,  (a)  Piles.  Piles  are  frequently  used  as  a 
foundation,  as  in  Fig.  76,  particularly  in  soft  ground,  and  also 
for  temporary  structures.  These  „ 

foundations   are    cheap,  quickly  con-         /  / 
structed,  and  are  particularly  valuable       /  / 
when  it  is  financially  necessary  to  open    ^~m — 
the  road  for  traffic  as  soon  as  possible   ^ppm^^p^ 

u     U 

FIG. 


and  with  the  least  expenditure  of 
money ;  but  there  is  the  disadvantage 
of  inevitable  decay  within  a  few  years  unless  the  piles  are  chemi- 
cally treated,  as  will  be  discussed  later.  Chemical  treatment, 
however,  increases  the  cost  so  that  such  a  foundation  would 
often  cost  more  than  a  foundation  of  stone.  A  pile  should  be 
driven  under  each  post  as  shown  in  Fig.  76. 


166 


RAILROAD   CONSTRUCTION. 


§140. 


(b)  Mud-sills. 

n  -\v/ 


WML 


Fig.  77  illustrates  the  use  of  mud-sills  as 
built  by  the  Louisville  and  Nash- 
ville R.  R.  Eight  blocks  12"  X  12" 
X  6'  are  used  under  each  bent. 
"When  the  ground  is  very  soft,  two 
M]  additional  timbers  (12"  X  12"  X 
length  of  bent-sill),  as  shown  by  the 


lp    dotted  lines,  are  placed  underneath. 
FIG.  77.  The  number  required  evidently  de- 

pends on  the  nature  of  the  ground. 

(c)  Stone  foundations.  Stone  foundations  are  the  best  and 
the  most  expensive.  For  very  high  trestles^  the  JSTorfolk  and 
Western  R.R.  employs  foundations  as  shown  in  Fig.  78,  the 


SILL    OF    TRESTLE 


FIG.  78. 

walls  being  4  feet  thick.  When  the  height  of  the  trestle  is  72 
feet  or  less  (the  plans  requiring  for  72'  in  height  a  foundation- 
wall  39'  6'  long)  the  foundation  is  made  continuous.  The  sill 
of  the  trestle  should  rest  on  several  short  lengths  of  3"  X  12" 
plank,  laid  transverse  to  the  sill  on  top  of  the  wall. 

140.  Longitudinal  bracing.  This  is  required  to  give  the 
structure  longitudinal  stiffness  and  also  to  reduce  the  columnar 
length  of  the  posts.  This  bracing  generally  consists  of  hori- 
zontal "  waling-strips "  and  diagonal  braces.  Sometimes  the 
braces  are  placed  wholly  on  the  outside  posts  unless  the  trestle 
is  very  high.  For  single-story  trestles  the  P.  R.  R.  employs 
the  "  laced  "  system,  i.e.,  a  line  of  posts  joining  the  cap  of  one 
bent  with  the  sill  of  the  next,  and  the  sill  of  that  bent  with  the 
cap  of  the  next.  Some  plans  employ  braces  forming  an  X  in 
alternate  panels.  Connecting  these  braces  in  the  center  more 
than  ddubles  their  columnar  strength.  Diagonal  braces,  when 
bolted  to  posts,  should  be  fastened  to  them  as  near  the  ends  of 


§  143.  TRESTLES.  167 

the  posts  as  possible.  The  sizes  employed  vary  largely,  depend- 
ing on  the  clear  length  and  on  whether  they  are  expected  to  act 
by  tension  or  compression.  3"  X  12"  planks  are  often  used 
when  the  design  would  require  tensile  strength  only,  and 
8"  X  8"  posts  are  often  used  when  compression  may  be 
expected. 

141.  Lateral  bracing.     Several  of  the  more  recent  designs  of 
trestles  employ  diagonal  lateral  bracing  between  the  caps  of 
adjacent  bents.     It  adds  greatly  to  the  stiffness  of  the  trestle 
and  better  maintains  its  alignment.      6"  X  6"  posts,   forming 
an  X  and  connected  at  the  center,  will  answer  the  purpose. 

142.  Abutments.     When    suitable   stone   for   masonry  is  at 
hand  and  a  suitable  subsoil  for  a  foundation  is  obtainable  without 
too  much   excavation,    a  masonry  abutment  will  be  the   best. 
Such    an   abutment   would    probably   be   used    when    masonry 
footings  for  trestle  bents  were  employed  (§  139,  <?). 

Another  method  is  to  construct  a  "  crib  "  of  10"  X  12" 
timber,  laid  horizontally,  drift-bolted  together,  securely  braced 
and  embedded  into  the  ground.  Except  for  temporary  con- 
struction such  a  method  is  generally  objectionable  on  account  of 
rapid  decay. 

Another  method,  used  most  commonly  for  pile  trestles,  and 
for  framed  .trestles  having  pile 
foundations  (§  139,  a),  is  to  use  a 
pile  bent  at  such  a  place  that  the 
natural  surface  on  the  up-hill  side 
is  not  far  below  the  cap,  and  the 
thrust  of  the  material,  filled  in  to 
bring  the  surface  to  grade,  is  insig- 
nificant. 3"  X  12"  planks  are  placed  FIG.  79. 

behind  the  piles,  cap,  and  stringers  to  retain  the  tilled  material. 

•*i^ 

FLOOR    SYSTEMS. 

143.  Stringers.     The  general  practice  is  to  use  two,  three, 
and  even  four  stringers  under  each  rail.     Sometimes  a  stringer 


168 


RAILROAD   CONSTRUCTION. 


§144 


is  placed  under  each  guard-rail.  Generally  the  stringers  are 
made  of  two  panel  lengths  and  laid  so  that  the  joints  alternate. 
A  few  roads  use  stringers  of  only  one  panel  length,  but  this 
practice  is  strongly  condemned  by  many  engineers.  The 
stringers  should  be  separated  to  allow  a  circulation  of  air  around 
them  and  prevent  the  decay  which  would  occur  if  they  were 
placed  close  together.  This  is  sometimes  done  by  means  of  2" 
planks,  6'  to  8'  long,  which  are  placed  over  each  trestle  bent. 
Several  bolts,  passing  through  all  the  stringers  forming  a  group 
and  through  the  separators,  bind  them  all  into  one  solid  con- 
struction. Cast-iron  u  spools  "  or  washers,  varying  from  4"  to 
£ "  in  length  (or  thickness),  are  sometimes  strung  on  each  bolt  so 
as  to  separate  the  stringers.  Sometimes  washers  are  used 
between  the  separating  planks  and  the  stringers,  the  object  of 
the  separating  planks  then  being  to  bind  the  stringers,  especially 
abutting  stringers,  and  increase  their  stiffness. 

The  most  common  size  for  stringers  is  8"  X  16".  The 
Pennsylvania  Railroad  varies  the  width,  depth,  and  number  of 
stringers  under  each  rail  according  to  the  clear  span.  It  may 
be  noticed  that,  assuming  a  uniform  load  per  running  foot,  both 


Clear  span. 

No.  of  pieces 
under  each  rail. 

Width. 

Depth. 

10  feet 

2 

8  iuches 

15  inches 

12    " 

2 

8      " 

16      " 

14    " 

2 

10      " 

17      " 

16    " 

3 

8      " 

17      " 

the  pressure  per  square  inch  at  the  ends  of  the  stringers  (the 
caps  having  a  width  of  12")  and  also  the  stress  due  to  trans- 
verse strain  are  kept  approximately  constant  for  the  variable  grog* 
load  on  these  varying  spans. 

144.  Corbels.  A  corbel  (in  trestle-work)  is  a  stick  of  timber 
(perhaps  two  placed  side  by  side),  about  3'  to  6'  long,  placed 
underneath  and  along  the  stringers  and  resting  on  the  cap. 
There  are  strong  prejudices  for  and  against  their  use,  and  a 


§  145.  TRESTLES.  169 

corresponding  diversity  in  practice.  They  are  bolted  to  the 
stringers  and  thus  stiffen  the  joint.  They  certainly  reduce  the 
objectionable  crushing  of  the  fibers  at  each  end  of  the  stringer, 
but  if  the  corbel  is  no  wider  than  the  stringers,  as  is  generally 
the  case,  the  area  of  pressure  between  the  corbels  and  the  cap  is 


FIG.  80. 

no  greater  and  the  pressure  per  square  inch  on  the  cap  is  no  less 
than  the  pressure  on  the  cap  if  no  corbels  were  used.  If  the 
corbels  and  cap  are  made  of  hard  wood,  as  is  recommended  by 
some,  the  danger  of  crushing  is  lessened,  but  the  extra  cost  and 
the  frequent  scarcity  of  hard  wood,  and  also  the  extra  cost  and 
labor  of  using,  corbels,  may  often  neutralize  the  advantages 
obtained  by  their  use. 

145.  Guard-rails.  These  are  frequently  made  of  5"  X  8" 
stuff,  notched  1"  for  each  tie.  The  sizes  vary  up  to  8"  X  8", 
and  the  depth  of  notch  from  f "  to  1£".  They  are  generally 
bolted  to  every  third  or  fourth  tie.  It  is  frequently  specified 
that  they  shall  be  made  of  oak,  white  pine,  or  yellow  pine.  The 
joints  are  made  over  a  tie,  by  halving  each  piece,  as  illustrated 
in  Fig.  81.  The  joints  on  opposite  sides  of  the  trestle  should  be 


FIG.  81. 

"  staggered."     Some  roads  fasten  every  tie  to  the  guard-rail, 
using  a  bolt,  a  spike,  or  a  lag-screw. 

Guard-rails  were  originally  used  with  the  idea  of  preventing 
the  wheels  of  a  derailed  truck  from  running  off  the  ends  of  the 
ties.  But  it  has  been  found  that  an  outer  guard-rail  alone  (with- 
out an  inner  guard-rail)  becomes  an  actual  element  of  danger, 
since  it  has  frequently  happened  that  a  derailed  wheel  has  caught 


170  RAILROAD  CONSTRUCTION.  §  147. 

on  the  outer  guard-rail,  thus  causing  the  truck  to  slew  around 
and  so  produce  a  dangerous  accident.  The  true  function  of  the 
outside  guard-rail  is  thus  changed  to  that  of  a  tie-spacer,  which 
keeps  the  ties  from  spreading  when  a  derailment  occurs.  The 
inside  guard-rail  generally  consists  of  an  ordinary  steel  rail 
spiked  about  10  inches  inside  of  the  running  rail.  These  inner 
guard-rails  should  be  bent  inward  to  a  point  in  the  center  of  the 
track  about  50  feet  from  the  end  of  the  bridge  or  trestle.  If 
the  inner  guard-rails  are  placed  with  a  clear  space  of  10  inches 
inside  the  running  rail,  the  outer  guard-rails  should  be  at  least 
&  10"  apart.  They  are  generally  much  farther  apart  than  this. 

146.  Ties  on  trestles.     If  a  car  is  derailed  on   a  bridge  or 
trestle,  the  heavily  loaded  wheels  are  apt  to  force  their  way  be- 
tween the  ties  by  displacing  them  unless  the  ties  are  closely 
spaced  and  fastened.     The  clear  space  between  ties  is  generally 
equal  to  or  less  than  their  width.     Occasionally  it  is  a  little  more 
than  their  width.     6"  X  8"  ties,  spaced  14"  to  16"  from  cen- 
ter to  center,  are  most  frequently  used.    The  length  varies  from 
9'  to  1 2'  for  single  track.     They  are  generally  notched  J"  deep 
on  the  under  side  where  they  rest  on  the  stringers.      Oak  ties 
are  generally  required  even  when  cheaper  ties  are  used  on  the 
other  sections  of  the  road.      Usually  every  third  or  fourth  tie  is 
bolted  to  the  stringers.     When  stringers  are  placed  underneath 
the  guard-rails,  bolts  are  run  from  the  top  of  the  guard-rail  to 
the  under  side  of  the  stringer.      The  guard-rails  thus  hold  down 
the  whole  system  of  ties,  and  no  direct  fastening  of  the  ties  to 
the  stringers  is  needed. 

147.  Superelevation  of  the  outer  rail  on  curves.     The  location 
of  curves  on  trestles   should  be  avoided  if  possible,  especially 
when    the  trestle  is  high.       Serious  additional  strains   are  in- 
troduced   especially    when    the    curvature    is    sharp     or    the 
speed  high.      Since  such  curves  are  sometimes  practically  un- 
;  avoidable,  it   is    necessary    to    design    the    trestle    accordingly. 
If  a  train  is  stopped  on  a  curved  trestle,  the  action  of  the  train  on 
the  trestle  is  evidently  vertical.     If  the  train  is  moving  with  a 
considerable  velocity,  the  resultant  of  the  weight  and  the  cen- 


§147. 


TRFSTLES. 


171 


trifugal  action  is  a  force  somewhat  inclined  from  the  vertical. 
Both  of  these  conditions  may  be  expected  to  exist  at  times.  If 
the  axis  of  the  system  of  posts  is  vertical  (as  illustrated  in 
methods  #,  5,  c,  d,  and  e),  any  lateral  force,  such  as  would  be 
produced  by  a  moving  train,  will  tend  to  rack  the  trestle  bent. 
If  the  stringers  are  set  vertically,  a  centrifugal  force  likewise 
tends  to  tip  them  sidewise.  If  the  axis  of  the  system  of  posts 
(or  of  the  stringers)  is  inclined  so  as  to  coincide  with  the  pressure 
of  the  train  on  the  trestle  when  the  train  is  moving  at  its  normal 
velocity,  there  is  no  tendency  to  rack  the  trestle  when  the  train 
is  moving  at  that  velocity,  but  there  will  be  a  tendency  to  rack 
the  trestle  or  twist  the  stringers  when  the  train  is  stationary. 
Since  a  moving  train  is  usually  the  normal  condition  of  affairs, 
as  well  as  the  condition  which  produces  the  maximum  stress,  an 
inclined  axis  is  evidently  preferable  from  a  theoretical  stand- 
point ;  but  whatever  design  is  adopted,  the  trestle  should  evi- 
dently be  sufficiently  cross-braced  for  either  a  moving  or  a 
stationary  load,  and  any  proposed  design  must  be  studied  as  to 
the  effect  of  both  of  these  conditions.  Some  of  the  various 
methods  of  securing  the  requisite  superelevation  may  be  described 
as  follows : 

(a)  Framing  the  outer  posts  longer  than  the  inner  posts,  so 
that  the  cap  is  inclined  at  the  proper  angle;  axis  of  posts  verti- 
cal.    (Fig.  82.)    The  method  requires 

more  work  in  framing  the  trestle, 
but  simplifies  subsequent  track-laying 
and  maintenance,  unless  it  should  be 
found  that  the  superelevation  adopted 
is  unsuitable,  in  which  case  it  could  be 
corrected  by  one  of  the  other  methods 
given  below.  The  stringers  tend  to 
twist  when  the  train  is  stationary.  PIG.  82 

(b)  Notching  the  cap  so  that  the  stringers  are  at  a  different 
elevation.    (Fig.  83.)     This  weakens  the  cap  and  requires  that 
all  ties  shall  be  notched  to  a  bevelled  surface  to  fit  the  stringers, 


172 


RAILROAD   CONSTRUCTION. 


§147. 


which  also  weakens  the  ties.     A  centrifugal  force  will  tend  to 

twist  the  stringers  and  rack  the  trestle, 
(c)  Placing  wedges  underneath  the 
ties  at  each  stringer.  These  wedges  are 
fastened  with  two  bolts.  Two  or  more 
wedges  will  be  required  for  each  tie. 
The  additional  number  of  pieces  re- 
quired for  a  long  curve  will  be  im- 
mense, and  the  work  of  inspection  and 
keeping  the  nuts  tight  will  greatly  in- 


FIG   83. 
crease  the  cost  of  maintenance. 

(d)  Placing  a  wedge  under  the  outer  rail  at  each  tie.     This 
requires  but  one  extra  piece  per  tie.     There  is  no  need  of  a 
wedge  under  the  inner  tie  in  prder  to  make  the  rail  normal  to 
the  tread.     The  resulting  inward  inclination,  is  substantially  that 
produced  by  some  forms  of  rail-chairs  or  tie-plates.    The  spikes 
(a  little  longer  than  usual)  are  driven  through  the  wedge  into 
the  tie.     Sometimes  "lag-screws"  are  used  instead  of  spikes. 
If  experience  proves  that  the  superelevation  is  too  much  or  too 
little,  it  may  be  changed  by  this  method  with  less  work  than  by 
any  other. 

(e)  Corbels  of  different  heights.     When  corbels  are  used  (see 
§  144)  the  required  inclination  of  the  floor  system  may  be  ob- 
tained by  varying  the  depth  of  the  corbels. 

(f )  Tipping  the  whole  trestle. 
This   is    done    by   placing   the 
trestle  on    an   inclined   founda- 
tion.    If   very  much    inclined, 
the  trestle  bent  must  be  secured 
against   the  possibility  of   slip- 
ping   sidewise,    for    the    slope 
would   be   considerable  with   a 
sharp  curve,  and  the  vibration 

of  a  moving  train  would  reduce  FIG.  84. 

the  coefficient  of  friction  to  a  comparatively  small  quantity, 
(g)   Framing   the   outer   posts  longer,      This  case  is  identical 


§  149.  TRESTLES.  173 

with,   case  (a)   except   that   the  axis  of   the  system  of  posts  is 
inclined,  as  in  case  (/*),  but  the  sill  is  horizontal. 

The  above-described  plans  will  suggest  a  great  variety  of 
methods  which  are  possible  and  which  differ  from  the  above 
only  in  minor  details. 

148.  Protection  from  fire.     Trestles  are  peculiarly  subject  to 
fire,  from  passing  locomotives,  which  may  not  only  destroy  the 
trestle,  but  perhaps  cause  a  terrible  disaster.     This  danger  is 
sometimes  reduced  by  placing  a  strip  of  galvanized  iron  along 
the  top  of  each  set  of  stringers  and  also  along  the  tops  of  the 
caps.     Still  greater  protection  was  given  on  a  long  trestle  on  the 
Louisville  and  Nashville  R.  R.   by  making  a  solid  flooring  of 
timber,  covered  with  a  layer  of  ballast  on  which  the  ties  and 
rails  were  laid  as  usual. 

Barrels  of  water  should  be  provided  and  kept  near  all  trestles, 
and  on  very  long  trestles  barrels  of  water  should  be  placed  every 
two  or  three  hundred  feet  along  its  length.  A  place  for  the  bar- 
rels may  be  provided  by  using  a  few  ties  which  have  an  extra 
length  of  about  four  feet,  thus  forming  a  small  platform,  which 
should  be  surrounded  by  a  railing.  The  track- walkers  should  be 
held  accountable  for  the  maintenance  of  a  supply  of  water  in 
these  barrels,  renewals  being  frequently  necessary  on  account  of 
evaporation.  Such  platforms  should  also  be  provided  as  REFUGE- 
BAYS  for  track- walkers  and  trackmen  working  on  the  trestle.  On 
very  long  trestles  such  a  platform  is  sometimes  provided  with 
sufficient  capacity  for  a  hand-car. 

149.  Timber.     Any  strong  durable  timber  may  be  used  when 
the  choice  is  limited,  but. oak,   pine,  or  cypress  are  preferred 
when  obtainable.      When  all  of  these  are  readily  obtainable, 
the  various  parts  of  the  trestle  will  be  constructed  of  different 
kinds  of  wood — the  stringers  of  long-leaf  pine,  the  posts  and 
braces  of  pine  or  red  cypress,  and  the  caps,  sills,  and  corbels  (if 
used)  of  white  oak.     The  use  of  oak  (or  a  similar  hard  wood) 
for  caps,  sills,  and  corbels  is   desirable  because  of  its  greater 
strength   in  resisting  crushing  across  the  grain,   which  is  the 
critical  test  for  these  parts.     There  is  no  physiological  basis  to 


174  RAILROAD  CONSTRUCTION.  §  151. 

the  objection,  sometimes  made,  that  different  species  of  timber, 
in  contact  with  each  other,  will  rot  quicker  than  if  only  one 
kind  of  timber  is  used.  When  a  very  extensive  trestle  is  to  be 
built  at  a  place  where  suitable  growing  timber  is  at  hand  but 
there  is  no  convenient  sawmill,  it  will  pay  to  transport  a  port- 
able sawmill  and  engine  and  cut  up  the  timber  as  desired. 

150.  Cost  of  framed  timber  trestles.     The  cost  varies  widely 
on  account  of  the  great  variation  in  the  cost  of  timber.     When 
a  railroad  is  first  penetrating  a  new  and  undeveloped  region,  the 
cost  of  timber  is  frequently  small,  and  when  it  is  obtainable  from 
the    company's    right-of-way  the  only  expense  is  felling  and 
sawing.     The  work  per  M.,  B.  M.,  is  small,  considering  that  a 
single  stick  12"  X  12"  X  25' contains  300  feet,  B.  M.,   and 
that    sometimes   a  few  hours'  work,  worth  less  than  $1,   will 
finish  all  the  work  required  on  it.     Smaller  pieces  will  of  course 
require  more  work  per  foot,  B.  M.     Long-leaf  pine  can  be  pur- 
chased from  the  mills  at  from  $8  to  $12  per  M.  feet,  B.  M., 
according  to  the  dimensions.     To  this  must  be  added  the  freight 
and  labor  of  erection.     The  cartage  from  the  nearest  railroad  to 
the  trestle  may  often  be  a  considerable  item.     Wrought  iron 
will  cost  about  3  c.  per  pound  and  cast  iron  2  c.,  although  the 
prices  are  often  lower  than  these.     The  amount  of  iron    used 
depends  on  the  detailed  design,  but,  as  an  average,  will  amount 
to  $1.50  to  $2  per  1000  feet,  B.  M.,  of  timber.    A  large  part  of 
the  trestling  of  the  country  has  been  built  at  a  contract  price  of 
about  $30  per  1000  feet,  B.  M.,  erected.     While  the  cost  will 
frequently  rise  to  $40  and  even  $50  when  timber  is  scarce,   it 
will  drop  to  $13  (cost  quoted)  when  timber  is  cheap. 

DESIGN  OF    WOODEN    TRESTLES. 

151.  Common  practice.     A  great  deal  of  trestling  has  been 
constructed  without  any  rational  design  except  that  custom  and 
experience  have  shown  that  certain  sizes  and  designs  are  probably 
safe.     This    method  has  resulted    occasionally  in   failures    but 
more  frequently  in  a  very  large  waste  of  timber.     Many  railroads 


§  152.  TRESTLES.  175 

employ  a  uniform  size  for  all  posts,  caps,  and  sills,  and  a 
uniform  size  for  stringers,  all  regardless  of  the  height  or  span  of 
the  trestle.  For  repair  work  there  are  practical  reasons  favoring 
this.  ' '  To  attempt  to  run  a  large  lot  of  sizes  would  be  more 
wasteful  in  the  end  than  to  maintain  a  few  stock  sizes  only. 
Lumber  can  be  bought  more  cheaply  by  giving  a  general  order 
for  'the  run  of  the  mill  for  the  season,'  or  'a  cargo  lot,' 
specifying  approximate  percentages  of  standard  stringer  size,  of 
12  X  12-inch  stuff,  10  X  10-inch  stuff,  etc.,  and  a  liberal  pro- 
portion of  3-  or  4-inch  plank,  all  lengths  thrown  in.  The  12  x  12- 
inch  stuff,  etc.,  is  ordered  all  lengths,  from  a  certain  specified 
length  up.  In  case  of  a  wreck,  washout,  burn-out,  or  sudden 
call  for  a  trestle  to  be  completed  in  a  stated  time,  it  is  much 
more  economical  and  practical  to  order  a  certain  number  of 
carloads  of  '  trestle  stuff '  to  the  ground  and  there  to  select  piece 
after  piece  as  fast  as  needed,  dependent  only  upon  the  length  of 
stick  required.  When  there  is  time  to  make  the  necessary  sur- 
veys of  the  ground  and  calculations  of  strength,-  and  to  wait  for  a 
special  bill  of  timber  to  be  cut  and  delivered,  the  use  of  differ- 
ent sizes  for  posts  in  a  structure  would  be  warranted  to  a  certain 
extent."*  For  new  construction,  when  there  is  generally 
sufficient  time  to  design  and  order  the  proper  sizes,  such  waste- 
fulness is  less  excusable,  and  under  any  conditions  it  is  both 
safer  and  more  economical  to  prepare  standard  designs  which 
can  be  made  applicable  to  varying  conditions  and  which  will  at 
the  same  time  utilize  as  much  of  the  strength  of  the  timber  as 
can  be  depended  on.  In  the  following  sections  will  be  given 
the  elements  of  the  preparation  of  such  standard  designs,  which 
will  utilize  uniform  sizes  with  as  little  waste  of  timber  as  possible. 
It  is  not  to  be  understood  that  special  designs  should  be  made 
for  each  individual  trestle. 

152.  Required  elements  of  strength.  The  stringers  of 
trestles  are  subject  to  transverse  strains,  to  crushing  across  the 
grain  at  the  ends,  and  to  shearing  along  the  neutral  axis.  The 

*  From  "Economical  Designing  of  Timber  Trestle  Bridges." 


176  RAILROAD  CONSTRUCTION.  §  153. 

strength  of  the  timber  must  therefore  be  computed  for  all  these 
kinds  of  stress.  Caps  arid  sills  will  fail,  if  at  all,  by  crushing 
across  the  grain ;  although  subject  to  other  forms  of  stress,  these 
could  hardly  cause  failure  in  the  sizes  usually  employed.  There 
is  an  apparent  exception  to  this :  if  piles  are  improperly  driven 
and  an  uneven  settlement  subsequently  occurs,  it  may  have  the 
effect  of  transferring  practically  all  of  the  weight  to  two  or  three 
piles,  while  the  cap  is  subjected  to  a  severe  transverse  strain 
which  may  cause  its  failure.  Since  such  action  is  caused  gener- 
ally by  avoidable  errors  of  construction  it  may  be  considered  as 
abnormal,  and  since  such  a  failure  will  generally  occur  by  a 
gradual  settlement,  all  danger  'may  be  avoided  by  reasonable 
care  in  inspection.  Posts  must  be  tested  for  their  columnar 
strength.  These  parts  form  the  bulk  of  the  trestle  and  are  the 
parts  which  can  be  definitely  designed  from  known  stresses. 
The  stresses  in  the  bracing  are  more  indefinite,  depending  on 
indeterminate  forces,  since  the  inclined  posts  take  up  an  un- 
known proportion  of  the  lateral  stresses,  and  the  design  of  the 
bracing  may  be  left  to  what  experience  has  shown  to  be  safe, 
without  involving  any  large  waste  of  timber. 

153.  Strength  of  timber.  Until  recently  tests  of  the  strength 
of  timber  have  generally  been  made  by  testing  small,  selected, 
well-seasoned  sticks  of  u  clear  stuff,"  free  from  knots  or  imper- 
fections. Such  tests  would  give  results  so  much  higher  than 
the  vaguely  known  strength  of  large  unseasoned  "  commercial  " 
timber  that  very  large  factors  of  safety  were  recommended — 
factors  so  large  as  to  detract  from  any  confidence  in  the  whole 
theoretical  design.  Recently  the  U.  S.  Government  has  been 
making  a  thoroughly  scientific  test  of  the  strength  of  full-size 
timber  under  various  conditions  as  to  seasoning,  etc.  The  work 
has  been  so  extensive  and  thorough  as  to  render  possible  the 
economical  designing  of  timber  structures. 

One  important  result  of  the  investigation  is  the  determina- 
tion of  the  great  influence  of  the  moisture  in  the  timber  and 
the  law  of  its  effect  on  the  strength.  It  has  been  also  shown 
that  timber  soaked  with  water  has  substantially  the  same 


153. 


TRESTLES. 


177 


strength  as  green  timber,  even  though  the  timber  had  once  been 
thoroughly  seasoned.  Since  trestles  are  exposed  to  the  weather 
they  should  be  designed  on  the  basis  of  using  green  timber. 
It  has  been  shown  that  the  strength  of  green  timber  is  very 
regularly  about  55  to  60$  of  the  strength  of  timber  in  which 
the  moisture  is  12$  of  the  dry  weight,  12$  being  the  proportion 
of  moisture  usually  found  in  timber  that  is  protected  from  the 
weather  but  not  heated,  as,  e.g.,  the  timber  in  a  barn.  Since 
the  moduli  of  rupture  have  all  been  reduced  to  this  standard  of 
moisture  (12$),  if  we  take  one-eighth  of  the  rupture  values,  it 
still  allows  a  factor  of  safety  of  about  five,  even  on  green  timber. 

Moduli  of  rupture  for  various  timbers.    [12£  moisture.] 
(Condensed  from  U.  S.  Forestry  Circular,  No.  15.) 


No. 

Species. 

Weight 
per 
cubic 
foot. 

Cross-bending. 

Crush- 
ing end 
wise. 

Crush- 
ing 
across 
grain. 

Shear- 
ing 
along 
grain. 

Ultimate 
Strength. 

Modulus  of 
Elasticity. 

1 

2 
3 
4 
5 
6 
7* 

Long-leaf  pine 
Cuban 
Short-leaf 

38 
39 
32 
33 
24 
31 
39 

12600 
13600 
10100 
11300 
7900 
9100 
10000 

2  070  000 
2  370  000 
1  680  000 
2  050  000 
1  390  000 
1  620  000 
1  640  000 

8000 
.8700 
6500 
7400 
5400 
6700 
7300 

1180 
1220 
960 
1150 
700 
1000 
1200 

700 
700 
700 
700 
400 
500 
800 

Loblolly 
White 
Red 
Spruce 

8 
9 
10 

Bald  cypress     .... 

29 
23 
32 

7900 
6300 
7900 

1290000 
910  000 
1  680  000 

6000 
5200 
5700 

800 
700 
800 

500 
400 
500 

White  cedar  

Douglas  spruce.  .  .  . 

11 
12 
13 
14 
15 
16 
19 
20 

White  oak         

50 
46 
50 
46 
45 
46 
45 
46 

13100 
11300 
12300 
11500 
11400 
13100 
10400 
12000 

2  090  000 
1  620  000 
2  030  000 
1  610  000 
1  970000 
1860000 
1  7M)  000 
1930000 

8500 
7300 
7100 
7400 
7200 
8100 
7200 
7700 

2200 
1900 
3000 
1900 
2300 
2000 
1600 
1800 

1000 
1000 
1100 
900 
1100 
900 
900 
900 

Overcup      

Post            

Cow             

Red              

Texan          .... 

Willow        
Spanish          

21 
27 
28 
29 
30 

Shagbark  hickory.  . 
Pignut            "       .  . 
White  elm     .    .  . 

51 
56 
34 
46 
39 

16000 
18700 
10300 
13500 
10800 

2390000 
2730000 
1540000 
1700000 
1640000 

9500 
10900 
6500 
8000 
7200 

2700 
3200 
1200 
2100 
1900 

1100 
1200 
800 
1300 
1100 

Cedar      "   

White  ash  

178 


RAILROAD  CONSTRUCTION. 


§1531 


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§  154.  TRESTLES.  179 

On  page  177  there  are  quoted  the  values  taken  from  the  U.  S. 
Government  reports  on  the  strength  of  timber,  the  tests  probably 
being  the  most  thorough  and  reliable  that  were  ever  made. 

On  page  178  are  given  the  "  average  safe  allowable  work- 
ino-  unit  stresses  in  pounds  per  square  inch,"  as  recommended 
by  the  committee  on  ' '  Strength  of  Bridge  and  Trestle  Timbers," 
the  work  being  done  under  the  auspices  of  the  Association  of 
Kailway  Superintendents  of  Bridges  and  Buildings.  The  report 
was  presented  at  their  fifth  annual  convention,  held  in  £s"ew 
Orleans,  in  October,  1895. 

154.  Loading.  As  shown  in  §  138,  the  span  of  trestles  is 
always  small,  is  generally  14  feet,  and  is  never  greater  than  18' 
except  when  supported  by  knee-braces.  The  greatest  load  that 
will  ever  come  on  any  one  span  will  be  the  concentrated  loading 
of  the  drivers  of  a  consolidation  locomotive.  With  spans  of  14 
feet  or  less  it  is  impossible  for  even  the  four  pairs  of  drivers  to 
be  on  the  same  span  at  once.  The  weight  of  the  rails,  ties,  and 
guard-rails  should  be  added  to  obtain  the  total  load  on  the  string- 
ers, and  the  weight  of  these,  plus  the  weight  of  the  stringers, 
should  be  added  to  obtain  the  pressure  on  the  caps  or  corbels. 
This  dead  load  is  almost  insignificant  compared  with  the  live 
load  and  may  be  included  with  it.  The  weight  of  rails,  ties, 
etc.,  may  be  estimated  at  200  pounds  per  foot.  To  obtain 
the  weight  on  the  caps  the  weight  of  the  stringers  must  be 
added,  which  depends  on  the  design  and  on  the  weight  per  cubic 
foot  of  the  wood  employed.  But  as  the  weight  of  the  stringers 
is  comparatively  small,  a  considerable  percentage  of  variation 
in  weight  will  have  but  an  insignificant  effect  on  the  result. 
Disregarding  all  refinements  as  to  actual  dimensions,  the  ordi- 
nary maximum  loading  for  standard  gauge  railroads  may  be 
taken  as  that  due  to  four  pairs  of  driving-axles,  spaced  5'  0" 
apart  and  giving  a  pressure  of  25,000  pounds  per  axle.  This 
should  be  increased  to  40,000  pounds  per  axle  (same  spacing) 
for  the  heaviest  traffic.  On  the  basis  of  25,000  pounds  per 
axle  the  following  results  have  been  computed : 


180 


EAILEOAD   CONSTRUCTION. 


§155. 


STRESSES    ON  VARIOUS    SPANS    DUE    TO    MOVING  LOADS  OF  25,000   POUNDS, 
SPACED  5'  0"   APART. 


Span  in  feet. 

Max.  mom.— 
ft.  Ibs. 

Max.  shear. 

Max  load  on 
one  cap. 

10 

65000 

38500 

52100 

12 

103  600 

45000 

62700 

14 

142  400 

49600 

74200 

16 

181400 

54725 

85700 

18 

220  600 

60100 

97900 

Although  the  dead  load  does  not  vary  in  proportion  to  the 
live  load,  yet,  considering  the  very  small  influence  of  the  dead 
load,  there  will  be  no  appreciable  error  in  assuming  the  corre- 
sponding values,  for  a  load  of  40,000  Ibs.  per  axle,  to  be  |f  of 
those  given  in  the  above  tabulation. 

155,  Factors  of  safety. — The  most  valuable  result  of  the  gov- 
ernment tests  is  the  knowledge  that  under  given  moisture  condi- 
tions the  strength  of  various  species  of  sound  timber  is  not  the 
variable  uncertain  quantity  it  was  once  supposed  to  be,  but  that 
its  strength  can  be  relied  on  to  a  comparatively  close  percentage. 
This  confidence  in  values  permits  the  employment  of  lower  fac- 
tors of  safety  than  have  heretofore  been  permissible.      Stresses, 
which   when   excessive  would  result  in  immediate  destruction, 
such  as  cross-breaking  and  columnar  stresses,  should  be  alloWed 
a  higher  factor  of  safety — say  6  or  8  for  green  timber.      Other 
stresses,  such  as  crushing  across  the  grain  and  shearing  along  the 
neutral  axis,  which  will  be  apparent  to  inspection  before  it  is 
dangerous,  may  be  allowed  lower  factors — say  3  to  5. 

156.  Design  of  stringers. — The  strength  of  rectangular  beams 
of  equal  width  varies  as  the  square  of  the  depth ;  therefore  deep 
beams  are  the  strongest.      On  the  other  hand,  when  any  cross- 
sectional    dimension  of    timber  much  exceeds  12"   the  cost  is 
much  higher  per  M.,  B.M.,  and  it  is  correspondingly  difficult  to 
obtain  thoroughly   sound  sticks,    free    from    wind-shakes,    etc. 
Wind-shakes  especially  affect  the  shearing  strength.     Also,  if 
the  required  transverse  strength  is  obtained  by  using  high  nar- 
row stringers,  the  area  of  pressure  between  the  stringers  and  the 


§  156.  TRESTLES.  181 

cap  may  become  so  small  as  to  induce  crushing  across  the  grain. 
This  is  a  very  common  defect  in  trestle  design.  As  already  in- 
dicated in  §  138,  the  span  should  vary  roughly  with  the  average 
height  of  the  trestle,  the  longer  spans  being  employed  when  the 
trestle  bents  are  very  high,  although  it  is  usual  to  employ  the 
same  span  throughout  any  one  trestle. 

To  illustrate,  if  we  select  a  span  of  14  feet,  the  load  on  one 
cap  will  be  74,200  Ibs.  If  the  stringers  and  cap  are  made  of 
long-leaf  yellow  pine,  which  require  the  closely  determined  value 
of  1180  Ibs.  per  square  inch  to  produce  a  crushing  amounting 
to  3#  of  the  height  on  timber  with  12$  moisture,  we  may  use 
200  Ibs.  per  square  inch  as  a  safe  pressure  even  for  green  tim- 
ber; this  will  require  371  square  inches  of  surface.  If  the  cap 
is  12"  wide,  this  will  require  a  width  of  31  inches,  or  say  2 
stringers  under  each  rail,  each  8  inches  wide.  For  rectangular 
beams 


Moment  = 

Using  for  1?  the  safe  value  1575  Ibs.  per  square  inch,  we  have 
142400  X  12  =  i  X  1575  X  32  X  A', 


from  which  h  =  15".  9.  If  desired,  the  width  may  be  increased 
to  9"  and  the  depth  correspondingly  reduced,  which  will  give 
similarly  h  =  14//.8,  or  say  15".  This  shows  that  two  beams, 
9"  X  15",  under  each  rail  will  stand  the  transverse  bending  and 
have  more  than  enough  area  for  crushing. 
The  shear  per  square  inch  will  equal 

3    total  shear        3         49600 

2  cross  section  =  2  4  .X  9  X  15  =  bs'  ?er  ^  mch' 

which  is  a  safe  value,  although  it  should  preferably  be  less. 
Hence  the  above  combination  of  dimensions  will  answer. 

The  deflection  should  be  computed  to  see  if  it  exceeds  the 


182  RAILROAD   CONSTRUCTION.  §157. 


somewhat  arbitrary  standard  of  ^-J-g-  of  the  span.     The  deflection 
for  uniform  loading  is 

§w 

A  = 


in  which  I  =  length  in  inches  ; 

W  =  total  load,  assumed  as  uniform  ; 

E  =  modulus  of  elasticity,  given  as  2,070,000  Ibs. 

per  sq.   in.   for  long-leaf  pine,   12$   dry,   and  assumed  to  be 
1,200,000  for  green  timber.      Then 

_  5  X  72800  X  168* 

'    32  X  36  X  15*  X  1200000  " 


so  that  the  calculated  deflection  is  well  within  the  limit.  Of 
course  the  loading  is  not  strictly  uniform,  but  even  with  a  lib- 
eral allowance  the  deflection  is  still  safe. 

For  the  heaviest  practice  (40000  Ibs.  per  axle)  these  stringer 
dimensions  must  be  correspondingly  increased. 

157.  Design  of  posts.  Four  posts  are  generally  used  for 
single-track  work.  The  inner  posts  are  usually  braced  by  the 
cross-braces,  so  that  their  columnar  strength  is  largely  increased  ; 
but  as  they  are  apt  to  get  more  than  their  share  of  work,  the  ad- 
vantage is  compensated  and  they  should  be  treated  as  unsupported 
columns  for  the  total  distance  between  cap  and  sill  in  simple 
bents,  or  for  the  height  of  stories  in  multiple-story  construction. 
The  caps  and  sills  are  assumed  to  have  a  width  of  12".  It 
facilitates  the  application  of  bracing  to  have  the  columns  of  the 
same  width  and  vary  the  other  dimension  as  required. 

Unfortunately  the  experimental  work  of  the  U.  S.  Govern- 
ment on  timber  testing  has  not  yet  progressed  far  enough  to 
establish  unquestionably  a  general  relation  between  the  strength 
of  long  columns  and  the  crushing  strength  of  short  blocks.  The 


§  157.  ,  TRESTLES.  183 

following  formula  has  been  suggested,  but  it  cannot  be  consid- 
ered as  established  : 

700  +  Uc 


700+15*+*' 

f  =  allowable  working  stress  per  sq.  in.  for  long  columns  ; 
F=         "  "  "       "     "     "     "    short  blocks; 

I 

G=d> 

I  =  length  of  column  in  inches  ; 

d  =  least  cross-sectional  dimension  in  inches. 

Enough  work  has  been  done  to  give  great  reliability  to  the  two 
following  formulae  for  white  pine  and  yellow  pine,  quoted  from 
Johnson's  "  Materials  of  Construction,"  p.  684  : 

1  /  Aa 
Working  load  per  sq.  in.  —p  =  1000  —  jhrj  ,  long-leaf  pine  ; 

"          "     "     "     "  =p=    600-  i-(|)',  white  pine; 

in  which     I  =  length  of  column  in  inches,     and 

h  =  least  cross-sectional  dimension  in  inches. 

The  frequent  practice  is  to  use  12"  X  12"  posts  for  all  tres- 
tles. If  we  substitute  in  the  above  formula  I  =  20'  =  240"  and 
h  =  12",  we  have  p  =  1000  -  i(-2T%°-)2  =  900  Ibs. 

900  X  144  =  129600  Ibs.,  the  working  load  for  each  post. 
This  is  more  than  the  total  load  on  one  trestle  bent  and  il- 
lustrates the  usual  great  waste  of  timber.  leaking  the  post 
8"  X  12"  and  calculating  similarly,  we  have  p  =  775,  and 
the  working  load  per  column  is  775  X  96  =  74400  Ibs.  As 
considerable  must  be  allowed  for  <  '  weathering,  '  '  which  destroys 
the  strength  of  the  outer  layers  of  the  wood,  and  also  for  the 
dynamic  effect  of  the  live  load,  8"  X  12"  may  not  be  too  great, 


184  RAILROAD   CONSTRUCTION.  §  158. 

but  it  is  certainly  a  safe  dimension.  12"  X  6"  would  possibly 
prove  amply  safe  in  practice.  One  method  of  allowing  for 
weathering  is  to  disregard  the  outer  half-inch  on  all  sides  of  the 
post,  i.  e. ,  to  calculate  the  strength  of  a  post  one  inch  smaller  in 
each  dimension  than  the  post  actually  employed.  On  this  basis 
an  8"  X  12"  X  20'  post,  computed  as  a  1"  x  11'  post,  would 
have  a  safe  columnar  strength  of  706  Ibs.  per  square  inch.  With 
an  area  of  77  square  inches,  this  gives  a  working  load  of  54362 
Ibs.  for  each  post,  or  217448  Ibs.  for  the  four  posts.  Consider- 
ing that  74200  Ibs.  is  the  maximum  load  on  one  cap  (14  feet 
span),  the  great  excess  of  strength  is  apparent. 

158.  Design  of  caps  and  sills.     The  stresses  in  caps  and  sills 
are  very  indefinite,  except  as  to  crushing  across  the  grain.     As 
the  stringers  are  placed  almost  directly  over  the  inner  posts,  and 
as  the  sills  are  supported  just  under  the  posts,  the  transverse 
stresses  are  almost  insignificant.     In   the  above  case  four  posts 
have  an  area  of  4  X  12"  X  8"  =  384  sq.  in.     The  total  load, 
74200  Ibs.,  will  then  give  a  pressure  of  193  pounds  per  square 
inch,    which  is  within  the  allowable  limit.     This  one  feature 
might  require  the  use  of  8"  X  12"  posts  rather  than  6"  X  12" 
posts,  for  the  smaller  posts,  although  probably  strong  enough  as- 
posts,  would  produce  an  objectionably  high  pressure. 

159.  Bracing.     Although   some  idea  of  the  stresses  in  the 
bracing  could  be  found  from  certain  assumptions  as  to  wind- 
pressure,  etc. ,  yet  it  would  probably  not  be  found  wise  to  de- 
crease, for  the  sake  of  economy,  the  dimensions  which  practice 
has  shown  to  be  sufficient  for  the  work.     The  economy  that 
would  be  possible  would  be  too  insignificant  to  justify  any  risk. 
Therefore  the  usual  dimensions,  given  in  §§  139  and  140,  should 
be  employed. 


CHAPTEK  Y. 

TUNNELS. 


SURVEYING. 

160.  Surface  surveys.  As  tunnels  are  always  dug  from  each 
end  and  frequently  from  one  or  more  intermediate  shafts,  it  is 
necessary  that  an  accurate  surface  survey  should  be  made 
between  the  two  ends.  As  the  natural  surface  in  a  locality 
where  a  tunnel  is  necessary  is  almost  invariably  very  steep  and 
rough,  it  requires  the  employment  of  unusually  refined  methods 
of  work  to  avoid  inaccuracies.  It  is  usual  to  run  a  line  on  the 
surface  that  will  be  at  every  point  vertically  over  the  center  line 
of  the  tunnel.  Tunnels  are  generally  made  straight  unless 
curves  are  absolutely  necessary,  as  curves  add"  greatly  to  the 
cost.  Fig.  85  represents  roughly  a  longitudinal  section  of  the 


•*•-— woo-— *t— - eeoe-— -j-----w9--- - — -eoee- — —-- seeo— 

FIG.  85  —SKETCH  OF  SECTION  OF  THE  HOOSAC  TUNNEL. 

Hoosac  Tunnel.  Permanent  stations  were  located  at  A,  B,  C, 
D,  E,  and  F^  and  stone  houses  were  built  at  A,  B,  C,  and  D. 
These  were  located  with  ordinary  field  transits  at  first,  and  then 
all  the  points  were  placed  as  nearly  as  possible  in  one  vertical 
plane  by  repeated  trials  and  minute  corrections,  using  a  very 
large  specially  constructed  transit.  The  stations  D  and  F  were 
necessary  because  E  and  A  were  invisible  from  C  and  B. 

185 


186  RAILROAD  CONSTRUCTION.  §  160. 

The  alignment  at  A  and  E  having  been  determined  with  great 
accuracy,  the  true  alignment  was  easily  carried  into  the  tunnel. 

The  relative  elevations  of  A  and  E  were  determined  with 
great  accuracy.  Steep  slopes  render  necessary  many  settings 
of  the  level  per  unit  of  horizontal  distance  and  require  that  the 
\  work  be  unusually  accurate  to  obtain  even  fair  accuracy  per 
unit  of  distance.  The  levels  are  usually  re-run  many  times 
until  the  probable  error  is  a  very  small  quantity. 

The  exact  horizontal  distance  between  the  two  ends  of  the 
tunnel  must  also  be  known,  especially  if  the  tunnel  is  on  a 
grade.  The  usual  steep  slopes  and  rough  topography  likewise 
render  accurate  horizontal  measurements  very  difficult.  Fre- 
quently when  the  slope  is  steep  the  measurement  is  best 
obtained  by  measuring  along  the  slope  and  allowing  for  grade. 
This  may  be  very  accurately  done  by  employing  two  tripods 
(level  or  transit  tripods  serve  the  purpose  very  well),  setting 
them  up  slightly  less  than  one  tape-length  apart  and  measuring 
between  horizontal  needles  set  in  wooden  blocks  inserted  in  the 
top  of  each  tripod.  The  elevation  of  each  needle  is  also 
observed.  The  true  horizontal  distance  between  two  successive 
positions  of  the  needles  then  equals  the  square  root  of  the 
difference  of  the  squares  of  the  inclined  distance  and  the  differ- 
ence of  elevation.  Such  measurements  will  probably  be  more 
accurate  than  those  made  by  attempting  to  hold  the  tape 
horizontal  and  plumbing  down  with  plumb-bobs,  because  (1) 
it  is  practically  difficult  to  hold  both  ends  of  the  tape  truly 
horizontal ;  (2)  on  steep  slopes  it  is  impossible  to  hold  the  down- 
hill end  of  a  100-foot  tape  (or  even  a  2  5 -foot  length)  on  a  level 
with  the  other  end,  and  the  great  increase  in  the  number  of 
applications  of  the  unit  of  measurement  very  greatly  increases 
the  probable  error  of  the  whole  measurement ;  (3)  the  vibrations 
of  a  plumb-bob  introduce  a  large  probability  of  error  in  trans- 
ferring the  measurement  from  the  elevated  end  of  the  tape  to 
the  ground,  and  the  increased  number  of  such  applications  of 
the  unit  of  measurement  still  further  .increases  the  probable 
error. 


§161.  ,      TUNNELS.  187 

161.  Surveying  down  a  shaft.  If  a  shaft  is  sunk,  as  at  $, 
Fig.  85,  and  it  is  desired  to  dig  out  the  tunnel  in  botli  directions 
from  the  foot  of  the  shaft  so  as  to  meet  the  headings  from  the 
outside,  it  is  necessary  to  know,  when  at  the  bottom  of  the 
shaft,  the  elevation,  alignment,  and  horizontal  distance  from 
each  end  of  the  tunnel. 

The  elevation  is  generally  carried  down  a  shaft  by  means  of 
a  steel  tape.  This  method  involves  the  least  number  of  appli- 
cations of  the  unit  of  measurement  and  greatly  increases  the 
accuracy  of  the  final  result. 

The  horizontal  distance  from  each  end  may  be  easily  trans- 
ferred down  the  shaft  by  means  of  a  plumb-bob,  using  some  of 
the  precautions  described  in  the  next  paragraph. 

To  transfer  the  alignment  from  the  surface  to  the  bottom  of 
a  shaft  requires  the  highest  skill  because  the  shaft  is  always 
small,  and  to  produce  a  line  perhaps  several  thousand  feet  long 
in  a  direction  given  by  two  points  6  or  8  feet  apart  requires 
that  the  two  points  must  be  determined  with  extreme  accuracy. 
The  eminently  successful  method  adopted  in  the  Hoosac  Tunnel 
will  be  briefly  described :  Two  beams  were  securely  fastened 
across  the  top  of  the  shaft  (1030  feet  deep),  the  beams  being 
placed  transversely  to  the  direction  of  the  tunnel  and  as  far 
apart  as  possible  and  yet  allow  plumb-lines,  hung  from  the 
intersection  of  each  beam  with  the  tunnel  center  line,  to  swing 
freely  at  the  bottom  of  the  shaft.  These  intersections  of  the 
beams  with  the  center  line  were  determined  by  averaging  the 
results  of  a  large  number  of  careful  observations  for  alignment. 
Two  fine  parallel  wires,  spaced  about  -fa"  apart,  were  then 
stretched  between  the  beams  so  that  the  center  line  of  the 
tunnel  bisected  at  all  points  the  space  between  the  wires. 
Plumb-bobs,  weighing  15  pounds,  were  suspended  by  fine  wires 
beside  each  cross-beam,  the  wires  passing  between  the  two 
parallel  alignment  wires  and  bisecting  the  space.  The  plumb- 
bobs  were  allowed  to  swing  in  pails  of  water  at  the  bottom. 
Drafts  of  air  up  the  shaft  required  the  construction  of  boxes 
surrounding  the  wires.  Even  these  precautions  did  not  suffice 


188  RAILROAD  CONSTRUCTION.  §  162. 

to  absolutely  prevent  vibration  of  the  wire  at  the  bottom 
through  a  very  small  arc.  The  mean  point  of  these  vibrations 
in  each  case  was  then  located  on  a  rigid  cross-beam  suitably 
placed  at  the  bottom  of  the  shaft  and  at  about  the  level  of  the 
roof  of  the  tunnel.  Short  plumb-lines  were  then  suspended 
from  these  points  whenever  desired;  a. transit  was  set  (by  trial) 
so  that  its  line  of  collimation  passed  through  both  plumb  lines 
.and  the  line  at  the  bottom  could  thus  be  prolonged. 

162.  Underground  surveys.  Survey  marks  are  frequently 
placed  on  the  timbering,  but  they  are  apt  to  prove  unreliable  on 
account  of  the  shifting  of  the  timbering  due  to  settlement  of  the 
surrounding  material.  They  should  never  be  placed  at  the  bottom 
of  the  tunnel  on  account  of  the  danger  of  being  disturbed  or 
covered  up.  Frequently  holes  are  drilled  in  the  roof  and  filled 
with  wooden  plugs  in  which  a  hook  is  screwed  exactly  on  line. 
Although  this  is  probably  the  safest  method,  even  these  plugs  are 
not  always  undisturbed,  as  the  material,  unless  very  hard,  will 
often  settle  slightly  as  the  excavation  proceeds.  "When  a  tunnel 
is  perfectly  straight  and  not  too  long,  alignment-points  may  be 
given  as  frequently  as  desired  from  permanent  stations  located 
outside  the  tunnel  where  they  are  not  liable  to  disturbance. 
This  has  been  accomplished  by  running  the  alignment  through 
the  upper  part  of  the  cross- section,  at 
one  s^e  °^  *ne  center,  where  it  is  out  of 
the  way  of  the  piles  of  masonry  material, 

'///Ml  **^\\^  \v$$£  debris,  etc.,  which  are  so  apt  to  choke 
up  the  lower  part  of  the  cross-section. 
The  position  of  this  line  relative  to  the 
cross-section  being  fixed,  the  alignment 
of  any  required  point  of  the  cross-section 
is  readily  found  by  means  of  a  light  frame 
or  template  with  a  fixed  target  located 
where  this  line  would  intersect  the  frame 
FIG.  86.  when  properly  placed.  A  level-bubble 

ion  the  frame  will  assist  in  setting  the  frame  in  its  proper  position. 
In  all  tunnel  surveying  the  cross-wires  must  be  illuminated 


§  163.  TUNNELS.  189 

fty  a  lantern,  and  the  object  sighted  at  must  also  be  illuminated. 
A  powerful  dark-lantern  with  the  opening  covered  with  ground 
glass  has  been  found  useful.  This  may  be  used  to  illuminate  a 
plumb-bob  string  or  a  very  fine  rod,  or  to  place  behind  a  brass 
plate  having  a  narrow  slit  in  it,  the  axis  of  the  slit  and  plate 
being  coincident  with  the  plumb-bob  string  by  which  it  is  hung. 

On  account  of  the  interference  to  the  surveying  caused  by 
the  work  of  construction  and  also  by  the  smoke  and  dust  in  the 
air  resulting  from  the  blasting,  it  is  generally  necessary  to  make 
the  surveys  at  times  when  construction  is  temporarily  sus- 
pended. 

163.  Accuracy  of  tunnel  surveying.  Apart  from  the  very 
natural  desire  to  do  surveying  which  shall  check  well,  there  is 
an  important  financial  side  to  accurate  tunnel  surveying.  If 
the  survey  lines  do  not  meet  as  desired  when  the  headings  come 
together,  it  may  be  found  necessary,  if  the  error  is  of  appreciable 
size,  to  introduce  a  slight  curve,  perhaps  even  a  reversed  curve, 
into  the  alignment,  and  it  is  even  conceivable  that  the  tunnel 
section  would  need  to  be  enlarged  somewhat  to  allow  for  these 
curves.  The  cost  of  these  changes  and  the  perpetual  annoyance 
due  to  an  enforced  and  undesirable  alteration  of  the  original 
design  will  justify  a  considerable  increase  in  the  expenses  of  the 
survey.  Considering  that  the  cost  of  surveys  is  usually  but  a 
small  fraction  of  the  total  cost  of  the  work,  an  increase  of  10  or 
even  20$  in  the  cost  of  the  surveys  will  mean  an  insignificant 
addition  to  the  total  cost  and  frequently,  if  not  generally,  it  will 
result  in  a  saving  of  many  times  the  increased  cost.  The 
accuracy  actually  attained  in  two  noted  American  tunnels  is 
given  as  follows  :  The  Musconetcong  tunnel  is  about  5000  feet 
long,  bored  through  a  mountain  400  feet  high.  The  error  of 
alignment  at  the  meeting  of  the  headings  was  0'.04,  error  of 
levels  0'.015,  error  of  distance  0'.52.  The  Hoosac  tunnel  is 
over  25,000  feet  long.  The  heading  from  the  east  end  met  the 
heading  from  the  central  shaft  at  a  point  11 274  feet  from  the 
east  end  and  1563  feet  from  the  shaft.  The  error  in  align- 
ment was  -£$  of  an  inch,  that  of  levels  "a  few  hundredths," 


190  RAILROAD  CONSTRUCTION.  §  164. 

error  of  distance  ' c  trifling. ' '  The  alignment,  corrected  at  the 
shaft,  was  carried  on  through  and  met  the  heading  from  the  west 
end  at  a  point  10138  feet  from  the  west  end  and  2056  feet  from 
the  shaft.  Here  the  error  of  alignment  was  -£$"  and  that  of 
levels  0.134ft. 


DESIGN. 

164.  Cross-sections.  Nearly  all  tunnels  have  cross-sections 
peculiar  to  themselves — all  varying  at  least  in  the  details.  The 
general  form,  of  a  great  many  tunnels  is  that  of  a  rectangle  sur- 
mounted by  a  semi-circle  or  semi- ellipse.  In  very  soft  material 
an  inverted  arch  is  necessary  along  the  bottom.  In  such  cases 
the  sides  will  generally  be  arched  instead  of  vertical.  The  sides 
are  frequently  battered.  With  very  long  tunnels,  several  forms 
of  cross-section  will  often  be  used  in  the  same  tunnel,  owing  to 
differences  in  the  material  encountered.  In  solid  rock,  which 
will  not  disintegrate  upon  exposure,  no  lining  is  required,  and 
the  cross  section  will  be  the  irregular  section  left  by  the  blasting, 
the  only  requirement  being  that  no  rock  shall  be  left  within  the 
required  cross-sectional  figure.  Farther  on,  in  the  same  tunnel, 
when  passing  through  some  very  soft  treacherous  material,  it 
may  be  necessary  to  put  in  a  full  arch  lining — top,  sides,  and  bot- 
tom— which  will  be  nearly  circular  in  cross-section.  For  an 
illustration  of  this  see  Figs.  87  and  88. 

The  width  of  tunnels  varies  as  greatly  as  the  designs.  Single- 
track  tunnels  generally  have  a  width  of  15  to  16  feet.  Occa- 
sionally they  have  been  built  14  feet  wide,  and  even  less,  and 
also  up  to  18  feet,  especially  when  on  curves.  24  to  26  feet  is 
the  most  common  width  for  double  track.  Many  double-track 
tunnels  are  only  22  feet  wide,  and  some  are  28  feet  wide.  The 
heights  are  generally  19  feet  for  single  track  and  20  to  22  feet 
for  double  track.  The  variations  from  these  figures  are  con- 
siderable. The  lower  limits  depend  on  the  cross-section  of  the 
rolling  stock,  with  an  indefinite  allowance  for  clearance  and  ven- 
tilation. Cross-sections  which  coincide  too  closely  with  what  is 


164. 


TUNNELS. 


191 


FIG.  87.— HOOSAC  TUNNEL.    SECTION  THROUGH  SOLID  ROCK. 


FIG.  88.— HOOSAC  TUNNEL     SECTION  THROUGH  SOFT  GROUND. 


192 


RAILROAD  CONSTRUCTION. 


§165. 


absolutely  required  for  clearance  are  objectionable,  because  any 
slight  settlement  of  the  lining  which  would  otherwise  be  harm- 
less would  then  become  troublesome  and  even  dangerous.  Figs. 
87,  88,  and  89  *  show  some  typical  cross-sections. 


3.70m.  _    3.70m.  :HL_ 


FIG.  89.— ST.  CLOUD  TUNNEL. 

165.  Grade.  A  grade  of  at  least  0.2$  is  needed  for  drainage. 
If  the  tunnel  is  at  the  summit  of  two  grades,  the  tunnel  grade 
should  be  practically  level,  with  an  allowance  for  drainage,  the 
actual  summit  being  perhaps  in  the  center  so  as  to  drain  both 
ways.  When  the  tunnel  forms  part  of  a  long  ascending  grade, 
it  is  advisable  to  reduce  the  grade  through  the  tunnel  unless  the 
tunnel  is  very  short.  The  additional  atmospheric  resistance  and 
the  decreased  adhesion  of  the  driver  wheels  on  the  damp  rails  in 
a  tunnel  will  cause  an  engine  to  work  very  hard  and  still  more 
rapidly  vitiate  the  atmosphere  until  the  accumulation  of  poison- 
ous gases  becomes  a  source  of  actual  danger  to  the  engineer  and 
fireman  of  the  locomotive  and  of  extreme  discomfort  to  the 
passengers.  If  the  nominal  ruling  grade  of  the  road  were 
maintained  through  a  tunnel,  the  maximum  resistance  would  be 

*  Drinker's  "Tunneling." 


PLATE  II, 


TUNNEL-TIMBERING— ENGLISH  SYSTEM  (a). 


TUNNEL-TIMBERING— ENGLISH  SYSTEM  (6). 
(To  face  page  192.) 


>t£^r4*> 

UNIVERSITY 


PLATE    III. 


TUNNEL-TIMBERING— ENGLISH  SYSTEM  (d). 
(To  face  page  192.) 


OF   THK 

UNIVERSITY 


§  167.  ,  TUNNELS.  193 

found  in  the  tunnel.     This  would  probably  cause  trains  to  stall 
there,  which  would  be  objectionable  and  perhaps  dangerous. 

166.  Lining.     It  is  a  characteristic  of  many  kinds  of  rock 
and  of  all  earthy  material  that,  although  they  may  be  self-sus- 
taining when  first  exposed  to  the  atmosphere,  they  rapidly  dis- 
integrate and  require  that  the  top  and  perhaps  the  sides  and 
even  the  bottom  shall  be  lined  to  prevent  caving  in.     In  this 
country,  when  timber  is  cheap,  it  is  occasionally  framed  as  an 
arch  and  used  as  the  permanent  lining,  but  masonry  is  always 
to  be  preferred.     Frequently  the  cross-section  is  made  extra 
large  so  that  a  masonry  lining  may  subsequently  be  placed  inside 
the  wooden  lining  and  thus  postpone  a  large  expense  until  the 
road  is  better  able  to  pay  for  the  work.     In  very  soft  unstable 
material,  like  quicksand,  an  arch  of  cut  stone  voussoirs  may  be 
necessary  to  withstand  the  pressure.     A  good  quality  of  brick  is 
occasionally  used  for  lining,  as  they  are  easily  handled  and  make 
good  masonry  if  the  pressure  is  not  excessive.     Only  the  best 
of  cement  mortar  should  be  used,  economy  in  this  feature  being 
the  worst  of  folly.      Of  course  the  excavation  must  include  the 
outside  line  of  the  lining.     Any  excavation  which  is  made  out- 
side of  this  line  (by  the  fall  of  earth  or  loose  rock  or  by  exces- 
sive blasting)  must  be  refilled  with  stone  well  packed  in.     Occa- 
sionally it  is  necessary  to  fill  these  spaces  with  concrete.     Of 
course  it  is  not  necessary  that  the  lining  be  uniform  throughout 
the  tunnel. 

167,  Shafts.     Shafts  are  variously  made  with  square,  rectan- 
gular,  elliptical,   and  circular  cross-sections.     The   rectangular 
cross-section,  with  the  longer  axis  parallel  with  the  tunnel,  is 
most  usually  employed.     Generally  ,the  shaft  is  directly  over  the 
center  of  the  tunnel,  but  that  always  implies  a  complicated  con- 
nection between  the  linings  of  the  tunnel  and  shaft,  provided 
such  linings  are  necessary.     It  is  easier  to  sink  a  shaft  near  to 
one  side  of  the  tunnel  and  make  an  opening  through  the  nearly 
vertical  side  of  the  tunnel.    Such  a  method  was  employed  in  the 
Church  Hill  Tunnel,  illustrated  in  Fig.  90.*     Fig.  91  f  shows 

*  Drinker's  "Tunneling." 

f  Rfciha,  "  Lehrbuch  der  Gesammten  Tunnelbaukunsi." 


194 


RAILROAD   CONSTRUCTION. 


§167. 


a  cross-section  for  a  large  main  shaft.  Many  shafts  have  been 
built  with  the  idea  of  being  left  open  permanently  for  ventila- 
tion and  have  therefore  been  elaborately  lined  with  masonry. 


FIG.  90.— CONNECTION  WITH  SHAFT,  CHURCH  HILL  TUNNEL. 


FIG.  91.— CROSS-SECTION.  LARGE  MAIN  SHAFT. 

The  general  consensus  of  opinion  now  appears  to  be  that  shafts 
are  worse  than  useless  for  ventilation ;  that  the  quick  passage  of 
a  train  through  the  tunnel  is  the  most  effective  ventilator ;  and 
that  shafts  only  tend  to  produce  cross-currents  and  are  ineffective 
to  clear  the  air.  In  consequence,  many  of  these  elaborately 
lined  shafts  have  been  permanently  closed,  and  the  more  recent 


PLATE  IV. 


TUNNEL-TIMBERING— FRENCH  SYSTEM  (a), 


TUNNEL-TIMBERING -FRENCH  SYSTEM  (6). 
(To  face  page  194.) 


PLATE  V. 


TUNNEL  TIMBERING — BELGIAN  SYSTEM  (a). 


TUNNEL-TIMBEKIXG— BELGIAN  SYSTEM  (6). 
(To  face  page  194.) 


§169.  TUNNELS.  195 

practice  is  to  close  up  a  shaft  as  soon  as  the  tunnel  is  completed. 
Shafts  always  form  drain  age- wells  for  the  material  they  pass 
through,  and  sometimes  to  such  an  extent  that  it  is  a  serious 
matter  to  dispose  of  the  water  that  collects  at  the  bottom, 
requiring  the  construction  of  large  and  expensive  drains. 

168.  Drains.     A  tunnel  will  almost  invariably  strike  veins  of 
water  which  will  promptly  begin  to  drain  into  the  tunnel  and 
not  only  cause  considerable  trouble  and  expense  during  construc- 
tion, but  necessitate  the  provision  of  permanent  drains  for  its 
perpetual  disposal.     These  drains  must  frequently  be  so  large  as 
to  appreciably  increase  the  required  cross- section  of  the  tunnel. 
Generally  a  small  open  gutter  on  each  side  will  suffice  for  this 
purpose,   but  in  double-track  tunnels  a  large  covered  drain  is 
often  built  between  the  tracks.     It  is  sometimes  necessary  to 
thoroughly  grout  the  outside  of  the  lining  so  that  water  will  not 
force  its  way  through  the  masonry  and  perhaps  injure  it,  but 
may  freely  drain  down  the  sides  and  pass  through  openings  in 
the  side  walls  near  their  base  into  the  gutters. 

CONSTRUCTION. 

169.  Headings.     The  methods  of  all  tunnel  excavation  de- 
pend on  the  general  principle  that  all  earthy  material,  except 
the  softest  of  liquid  mud  and  quicksand,  will  be  self-sustaining 
over  a  greater  or  less  area  and  for'  a  greater  or  less  time  after 
excavation  is  made,  and  the  work  consists  in  excavating  some 
material  and  immediately  propping  up  the  exposed  surface  by 
timbering  and  poling-boards.     The  excavation  of  the  cross-sec- 
tion  begins  with  cutting  out  a  " heading,"   which  is  a  small 
horizontal   drift  whose   breast   is    constantly  kept    15    feet    or 
more  in  advance  of  the  full  cross-sectional  excavation.     In  solid  \ 
self-sustaining  rock,  which  will  not  decompose  upon  exposure 
to  aic,  it  becomes  simply  a  matter  of  excavating  the  rock  with 
the   least   possible   expenditure  of  time  and   energy.     In  soft 
ground  the  heading  must  be  heavily  timbered,  and  as  the  heading 
is  gradually  enlarged  the  timbering  must  be  gradually  extended 


196 


RAILROAD   CONSTRUCTION. 


§170. 


and  perhaps  replaced,  according  to  some  regular  system,  so  that 
when  the  full  cross- section  has  been  excavated  it  is  supported 
by  such  timbering  as  is  intended  for  it.  The  heading  is  some- 
times made  on  the  center  line  near  the  top ;  with  other  plans, 
on  the  center  line  near  the  bottom ;  and 
sometimes  two  simultaneous  headings  are  run 
in  the  two  lower  corners.  Headings  near  the 
bottom  serve  the  purpose  of  draining  the 
material  above  it  and  facilitating  the  excava- 
tion. The  simplest  case  of  heading  timber- 
ing is  that  shown  in  Fig.  92,  in  which  cross- 
timbers  are  placed  at  intervals  just  under  the 
roof,  set  in  notches  cut  in  the  side  walls  and 
supporting  poling-boards  which  sustain  what- 
ever pressure  may  come  on  them.  Cross-timbers  near  the  bottom 
support  a  flooring  on  which  vehicles  for  transporting  material 
may  be  run  and  under  which  the  drainage  may  freely  escape. 
As  the  necessity  for  timbering  becomes  greater,  side  timbers  and 
even  bottom  timbers  must  be  added,  these  timbers  supporting 
poling-boards,  and  even  the  breast  of  the  heading  must  be  pro- 
tected by  boards  suitably  braced,  as  shown  in  Fig.  93.  The 


FIG.  92. 


93  —  TIMBERING  FOR  TUNNEL  HEADING. 


supporting  timbers  are  framed  into  collars  in  such  a  manner  that 
added  pressure  only  increases  their  rigidity. 

170.  Enlargement.     Enlargement  is  accomplished  by  remov- 
ing the  poling-boards,  one  at  a  time,  excavating  a  greater  or  less 


PLATE  VI. 


TUNNEL  TIMBERING — GERMAN  SYSTEM  (a). 


•-%; 

•'V^- 


IgE K4£=^ry-  /;:     ; 

•       v  ^^~         '  [^ 

^  .  ;^x,--       — '  - 


TUNNEL-TIMBERING— GERMAN  SYS'JEM  (6). 
(To  face  page  196.) 


PLATE  VII. 


TUNNEL-TIMBERING— GERMAN  SYSTEM 


TUNNEL-TIMBERING—GERMAN  SYSTEM 
(To  face  page  196  ) 


§171. 


TUNNELS. 


197 


amount  of  material,  and  immediately  supporting  the  exposed 
material  with  poling-boards  suitably  braced.  (See  Figs.  93  and 
94.)  This  work  being  systematically  done,  space  is  thereby 


FIG.  94. 

obtained  in  which  the  framing  for  the  full  cross- section  may  be 
gradually  introduced.  The  framing  is  constructed  with  a  cross- 
section  so  large  that  the  masonry  lining  may  be  constructed 
within  it. 

171.  Distinctive  features  of  various  methods  of  construction. 
There  are  six  general  systems,  known  as  the  English,  German, 
Belgian,  French,  Austrian,  and  American.  They  are  so  named 
from  the  origin  of  the  methods,  although  their  use  is  not  confined 
to  the  countries  named.  Fig.  95  shows  by  numbers  (1  to  5) 
the  order  of  the  excavation  within  the  cross-sections.  The  Eng- 
lish, Austrian,  and  American  systems  are  alike  in  excavating  the 
entire  cross-section  before  beginning  the  construction  of  the 
masonry  lining.  The  German  method  leaves  a  solid  core  (5) 
until  practically  the  whole  of  the  lining  is  complete.  This  has 
the  disadvantage  of  extremely  cramped  quarters  for  work,  poor 
ventilation,  etc.  The  Belgian  and  French  methods  agree  in 
excavating  the  upper  part  of  the  section,  building  the  arch  at 
once,  and  supporting  it  temporarily  until  the  side  walls  are 
built.  The  Belgian  method  then  takes  out  the  core  (3),  removes 
very  short  sections  of  the  sides  (4),  immediately  underpinning 
the  arch  with  short  sections  of  the  side  walls  and  thus  gradually 
constructing  the  whole  side  wall.  The  French  method  digs  out 
the  sides  (3),  supporting  the  arch  temporarily  with  timbers  and 


198 


RAILROAD  CONSTRUCTION. 


§171. 


then  replacing  the  timbers  with  masonry ;  the  core  (4)  is  taken 
out  last.  The  French  method  has  the  same  disadvantage  as  the 
German — working  in  a  cramped  space.  The  Belgian  and  French 
systems  have  the  disadvantage  that  the  arch,  supported  tempo- 
rarily on  timber,  is  very  apt  to  be  strained  and  cracked  by  the 
slight  settlement  that  so  frequently  occurs  in  soft  material.  The 
English,  Austrian,  and  American  methods  differ  mainly  in  the 


4 

'         4    ' 

5  j 

1            5 



2 

4 

2 





1 

1 

AUSTRIAN 


AMERICAN 


!  3 


GERMAN  BELGIAN  FRENCH 

FIG.  95. — ORDER  OF  WORKING  BY  THE  VARIOUS  SYSTEMS. 

design  of  the  timbering.  The  English  support  the  roof  by  lines 
of  very  heavy  longitudinal  timbers  which  are  supported  at  com- 
paratively wide  intervals  by  a  heavy  framework  occupying  the 
whole  cross-section.  The  Austrian  system  uses  such  frequent 
cross-frames  of  timber-work  that  poling-boards  will  suffice  to 
support  the  material  between  the  frames.  The  American  sys- 
tem agrees  with  the  Austrian  in  using  frequent  cross-frames 
supporting  poling-boards,  but  differs  from  it  in  that  the  u  cross- 
frames  "  consist  simply  of  arches  of  3  to  15  wooden  voussoirs, 
the  voussoirs  being  blocks  of  12"  X  12"  timber  about  2  to  8  feet 
long  and  cut  with  joints  normal  to  the  arch.  These  arches  are 
put  together  on  a  centering  which  is  removed  as  soon  as  the  arch 


PLATE   VIII, 


TUNNEL-TIMBERING — AUSTRIAN    SYSTEM    (a). 


/;•  X"~^X    •' 
TUNNEL- TIMBERING — AUSTRIAN  SYSTEM  (6). 


TUNNEL-TIMBERING — AUSTRIAN  SYSTEM  (c). 
(To  face  page  193.) 


PLATE   IX. 


TUNNEL-TIMBEBING— AUSTRIAN  SYSTEM  (d). 


TUNNEL-TIMBERING — AUSTRIAN  SYSTEM  (e). 
(To  face  page  198.) 


PLATE   X, 


TUNNEL-TIMBERING—AUSTRIAN  SYSTEM  (/). 


TUNNEL-TIMBERING— AUSTRIAN  SYSTEM  (g). 
(To  face  page  198  ) 


$  173.  TUNNELS.  199 

is  keyed  up  and  thus  immediately  opens  up  the  full  cross-section, 
so  that  the  center  core  (4)  may  be  immediately  dug  out  and  the 
masonry  constructed  in  a  large  open  space.  The  American  sys- 
tem has  been  used  successfully  in  very  soft  ground,  but  its  ad- 
vantages are  greater  in  loose  rock,  when  it  is  much  cheaper  than 
the  other  methods  which  employ  more  timber.  Fig.  90  illus- 
trates the  use  of  the  American  system.  The  figure  shows  the 
wooden  arch  in  place.  The  masonry  arch  may  be  placed  when 
convenient,  since  it  is  possible  to  lay  the  track  and  commence 
traffic  as  soon  as  the  wooden  arch  is  in  place.  Plates  II  to  XIV 
illustrate  the  methods  of  excavating  and  timbering  by  these 
various  systems. 

172.  Ventilation  during  construction.     Tunnels  of  any  great 
length  must  be  artificially  ventilated  during  construction.     If 
the  excavated  material  is  rock  so  that  blasting  is  necessary,  the 
need  for  ventilation  becomes  still  more  imperative.      The  inven- 
tion of  compressed-air  drills  simultaneously  solved  two  difficul- 
ties.     It  introduced  a  motive  power  which  is  unobjectionable  in 
its  application  (as  gas  would  be),  and  it  also  furnished  at  the  same 
time  a  supply  of  just  what  is  needed — pure  air.      If  no  blasting 
is  done  (and  sometimes  even  when  there  is  blasting),  air  must  be 
supplied  by  direct  pumping.     The  cooling  effect  of  the  sudden 
expansion  of  compressed  air  only  reduces  the  otherwise  objection- 
ably high  temperature  sometimes  found  in  tunnels.     Since  pure 
air  is  being  continually  pumped  in,  the  foul  air  is  thereby  forced 
out. 

173.  Excavation  for  the  portals.     Under  normal  conditions 
there  is  always  a  greater  or  less  amount  of  open  cut   preceding 
and  following  a  tunnel.      Since  all  tunnel  methods  depend  (to 
some  slight  degree  at  least)  on  the  capacity  of  the  exposed  ma- 
terial to  act  as  an  arch,  there  is  implied  a  considerable  thickness 
of  material  above  the   tunnel.      TJiis  thickness   is  reduced   to 
nearly  zero  over  the  tunnel  portals  and  therefore  requires  special 
treatment,  particularly  when  the  material  is  very  soft.    Fig.  96  * 

*  Rziha,  "  Lehrbuch  der  Qersammten  Tunnelbaukunst." 


200 


RAILROAD  CONSTRUCTION. 


174. 


illustrates  one  method  of  breaking  into  the  ground  at  a  portal. 
The  loose  stones  are  piled  on  the  framing  to  give  stability  to  the 
framing  by  their  weight  and  also  to  retain  the  earth  on  the 
slope  above.  Another  method  is  to  sink  a  temporary  shaft  to 
the  tunnel  near  the  portal ;  immediately  enlarge  to  the  full  size 
and  build  the  masonry  lining ;  then  work  back  to  the  portal. 


FIG.  96.— TIMBERING  FOB  TUNNEL  PORTAL. 

This  method  is  more  costly,  but  is  preferable  in  very  treacherous 
ground,  it  being  less  liable  to  cause  landslides  of  the  surface 
material. 

174,  Tunnels  vs,  open  cuts.  In  cases  in  which  an  open  cut 
rather  than  a  tunnel  is  a  possibility  the  ultimate  consideration 
is  generally  that  of  first  cost  combined  with  other  financial  con- 
siderations and  annual  maintenance  charges  directly  or  indirectly 
connected  with  it.  Even  when  an  open  cut  may  be  constructed  at 
the  same  cost  as  a  tunnel  (or  perhaps  a  little  cheaper)  the  tunnel 
may  be  preferable  under  the  following  conditions : 

1 .  When  the  soil  indicates  that  the  open  cut  would  be  liable 
to  landslides. 

2.  When  the  open  cut  would  be  subject  to  excessive  snow- 
drifts or  avalanches. 

3.  When  land  is  especially  costly  or  it  is  desired  to  run  under 
existing  costly  or  valuable  buildings  or  monuments.     When  run- 
ning through  cities,  tunnels  are  sometimes  constructed  as  open 
cuts  and  then  arched  over. 


PLATE  XI, 


TEMPORARY  TIMBERING  OF  HEADING 


L— J 


PERMANENT  TIMBERING  OF  HEADING. 


*/         '  ~      "        "     i j 

PH(ENIXVILLE  TUNNEL.      P.    S.    V.    R.R. 

(To  face  page  200.) 


PLATE  XII. 


TIMBERING  OF  FULL  TUNNEL  SECTION 


PHCEKIXVILLE  TUNNEL.    P.  S.  V.  R.R. 
(To  face  page  200.) 


PLATE   XIII. 


PHCENIXVILLE  TUNNEL.    P.  S.  V.  R.R. 


(To  face  page  200.) 


PLATE  XIY. 


tUffr^^  =1=  =q^ 
iWZ7*o±ni5c±/JX, 


y 

VX 

/ 

-tt'-S1— 

Y 

V 

I                               V 

irf*          -t3  '-e-'  

LONGITUDINAL  SECTION  OF  PORTAL. 
PHCENIXVILLE  TUNNEL.    P.  S.  V.  R.R. 


§175. 


TUNNELS. 


201 


These  cases  apply  to  tunnels  vs.  open  cuts  when  the  align- 
ment is  fixed  by  other  considerations  than  the  mere  topography. 
The  broader  question  of  excavating  tunnels  to  avoid  excessive 
grades  or  to  save  distance  or  curvature,  and  similar  problems, 
are  hardly  susceptible  of  general  analysis  except  as  questions  of 
railway  economics  and  must  be  treated  individually. 

175.  Cost  of  tunneling.  The  cost  of  any  construction  which 
involves  such  uncertainties  as  tunneling  is  very  variable.  It  de- 
pends on  the  material  encountered,  the  amount  and  kind  of  tim- 
bering required,  on  the  size  of  the  cross- section,  on  the  price  of 
labor,  and  especially  on  the  reconstruction  that  may  be  necessary 
on  account  of  mishaps. 

Headings  generally  cost  $4  to  $5  per  cubic  yard  for  excava- 
tion, while  the  remainder  of  the  cross-section  in  the  same  tunnel 
may  cost  about  half  as  much.  The  average  cost  of  a  large  number 
of  tunnels  in  this  country  may  be  seen  from  the  following  table :  * 


Material. 

Cost  per  cubic  yard. 

Cost  per 
lineal  foot. 

Excavation. 

Masonry. 

Single. 

Double. 

Single. 

Double. 

Single. 

Double. 

Hard  rock  

$5.89 

$5.45 

$12.00 

$  8.25 

$  69.76 

$142.82 

Loose  rock  

3.12 

3.48 

9.07 

10.41 

80.61 

119.26 

Soft  ground  .  .  . 

3.62 

4.64 

15.00 

10.50 

135.31 

174.42 

A  considerable  variation  from  these  figures  may  be  found  in 
individual  cases,  due  sometimes  to  unusual  skill  (or  the  lack  of 
it)  in  prosecuting  the  work,  but  the  figures  will  generally  be 
sufficiently  accurate  for  preliminary  estimates  or  for  the  compari- 
son of  two  proposed  routes. 


*  Figures  derived  from  Drinker's  "Tunneling." 


CHAPTEK  VI. 

CULVERTS  AND  MINOR  BRIDGES. 

176.  Definition  and  object.     Although  a  variable  percentage 
of  the  rain  falling  on  any  section  of  country  soaks  into  the 
ground  and  does  not  immediately  reappear,  yet  a  very  large 
percentage  flows  over  the  surface,  always  seeking  and  following 
the  lowest  channels.      The  roadbed  of  a  railroad  is  constantly 
intersecting  these  channels,  which  frequently  are  normally  dry. 
In  order  to  prevent  injury  to  railroad  embankments  by  the  im- 
pounding of  such  rainfall,  it  is  necessary  to  construct  waterways 
-through  the   embankment  through  which    such   rairiflow   may 
freely  pass.     Such  waterways,   called  culverts,  are  also  appli- 
cable for  the  bridging  of  very  small  although  perennial  streams, 
and  therefore  in  this  work  the  term  culvert  will  be  applied  to 
all  water- channels  passing  through  a  railroad  embankment  which 
are  not  of  sufficient  magnitude  to  require  a  special  structural 
design,  such  as  is  necessary  for  a  large  masonry  arch  or  a  truss 
bridge. 

177.  Elements  of  the  design.     A  well-designed  culvert  must 
afford  such  free  passage  to  the  water  that  it  will  not  ' '  back  up ' ' 
over  the  adjoining  land  nor  cause  any  injury  to  the  embankment 
or  culvert.     The  ability  of  the  culvert  to  discharge  freely  all  the 
water  that  comes  to  it  evidently  depends  chiefly  on  the  area  of 
the  waterway,  but  also  on  the  form,  length,  slope,  and  materials 
of  construction  of  the   culvert  and  the  nature  of  the  approach 
and  outfall.     When  the  embankment  is  very  low  and  the  amount 
of  water  to  be  discharged  very  great,  it   sometimes  becomes 
necessary  to  allow  the  water  to  discharge  "  under  a  head,"  i.e., 

202 


§  178.  CULVERTS  .AND  MINOR  BRIDGES.  203 

with  the  surface  of  the  water  above  the  top  of  the  culvert. 
Safety  then  requires  a  much  stronger  construction  than  would 
otherwise  be  necessary  to  avoid  injury  to  the  culvert  or  embank- 
ment by  washing.  The  necessity  for  such  construction  should 
be  avoided  if  possible. 

AREA    OF    THE    WATERWAY. 

178.  Elements  involved.  The  determination  of  the  required 
area  of  the  waterway  involves  such  a  multiplicity  of  indeter- 
minate elements  that  any  close  determination  of  its  value  from 
purely  theoretical  considerations  is  a  practical  impossibility. 
The  principal  elements  involved  are: 

a.  Rainfall.     The  real  test  of  the  culvert  is  its  capacity  to 
discharge  without  injury  the  flow  resulting  from  the  extraordi- 
nary rainfalls  and  "cloud  bursts"  that  may  occur  once  in  many 
years.      Therefore,    while  a   knowledge  of  the  average  annual 
rainfall  is  of  very  little  value,  a  record  of  the  maximum  rainfall 
during  heavy  storms  for  a  long  term  of  years  may  give  a  relative 
idea  of  the  maximum  demand  on  the  culvert. 

b.  Area  of  watershed.    This  signifies  the  total  area  of  country 
draining   into   the    channel    considered.       When   the   drainage 
area   is  very  small    it    is  sometimes  included   within   the  area 
surveyed  by  the  preliminary    survey.      When  larger   it  is  fre- 
quently possible  to  obtain  its  area  from  other  maps  with  a  per- 
centage of  accuracy  sufficient  for  the   purpose.      Sometimes  a 
special  survey  for  the  purpose  is  considered  justifiable. 

c.  Character   of  soil   and  vegetation,      This  has  a  large  in- 
fluence on  the  rapidity  with  which  the  rainflow  from  a  given 
area  will  reach  the  culvert.     If  the  soil  is  hard  and  impermeable 
and  the  vegetation  scant,  a  heavy  rayi  will  run  off  suddenly, 
taxing   the   capacity  of   the   culvert   for  a  short  time,   while  a 
spongy  soil  and  dense  vegetation  will  retard  the  flow,  making  it 
more  nearly  uniform  and  the  maximum  flow  at  any  one  time 
much  less. 

d.  Shape   and  slope  of  watershed.     If  the  watershed  is  very 
long  and  narrow  (other  things  being  equal),  the  water  from  the 


204  RAILROAD  CONSTRUCTION.  §179. 

remoter  parts  will  require  so  much  longer  time  to  reach  the 
culvert  that  the  flow  will  be  comparatively  uniform,  especially 
when  the  slope  of  the  whole  watershed  is  very  low.  When  the 
slope  of  the  remoter  portions  is  quite  steep  it  may  result  in  the 
nearly  simultaneous  arrival  of  a  storm-flow  from  all  parts  of  the 
watershed,  thus  taxing  the  capacity  of  the  culvert. 

e.  Effect  of  design  of  culvert.  The  principles  of  hydraulics 
show  that  the  slope  of  the  culvert,  its  length,  the  form  of  the 
cross-section,  the  nature  of  the  surface,  and  the  form  of  the 
approach  and  discharge  all  have  a  considerable  influence  on  the 
area  of  cross -section  required  to  discharge  a  given  volume  of 
water  in  a  given  time,  but  unfortunately  the  combined 
hydraulic  effect  of  these  various  details  is  still  a  very  uncertain 
quantity. 

179,  Methods  of  computation  of  area.  There  are  three  pos- 
sible methods  of  computation. 

(a)  Theoretical,     As  shown  above  it  is  a  practical  impossi- 
bility to  estimate  correctly  the  combined  effect  of  the  great  mul- 
tiplicity of  elements  which  influence  the  final  result.    The  nearest 
approach  to  it  is  to  estimate  by  the  use  of  empirical  formulae 
the  amount  of  water  which  will  be  presented  at  the  upper  end 
of  the  culvert  in  a  given  time  and  then  to  compute,  from  the 
principles  of  hydraulics,  the  rate  of  flow  through  a  culvert  of 
given  construction,  but  (as  shown  in  §  178,  e)  such  methods  are 
still  very  unreliable,  owing  to  lack  of  experimental  knowledge. 
This  method  has  apparently  greater  scientific  accuracy  than  other 
methods,  but  a  little  study  will  show  that  the  elements  of  un- 
certainty are  as  great  and  the  final  result  no  more  reliable.     The 
method  is  most  reliable  for  streams  of  uniform  flow,  but  it  is 
under  these  conditions  that  method  (c)  is   most  useful.     The 
theoretical  method  will  not  therefore  be  considered  further. 

(b)  Empirical.     As  illustrated  in  §  180,  some  formulae  make 
the  area  of  waterway  a  function  of  the  drainage  area,  the  for- 
mula being  affected  by  a  coefficient  the  value  of  which  is  esti- 
mated between  limits  according  to  the  judgment.     Assuming 
that  the  formulae  are  sound,  their  use  only  narrows  the  limits  of 


§180.  CULVERTS^ AND  MINOR  BRIDGES.  205 

error,  the  final  determination  depending  on  experience  and  judg- 
ment. 

(c)  From  observation.  This  method,  considered  by  far  the 
best  for  permanent  work,  consists  in  observing  the  high-water 
marks  on  contracted  channel-openings  which  are  on  the  same 
stream  and  as  near  as  possible  to  the  proposed  culvert.  If  the 
country  is  new  and  there  are  no  such  openings,  the  wisest  plan 
is  to  bridge  the  opening  by  a  temporary  structure  in  wood  which 
has  an  ample  waterway  (see  §  126,  5,  4)  and  carefully  observe 
all  high- water  marks  on  that  opening  during  the  6  to  10  years 
which  is  ordinarily  the  minimum  life  of  such  a  structure.  As 
shown  later,  such  observations  may  be  utilized  for  a  close  com- 
putation of  the  required  waterway.  Method  (b)  may  be  utilized 
for  an  approximate  calculation  for  the  required  area  for  the  tem- 
porary structure,  using  a  value  which  is  intentionally  excessive, 
so  that  a  permanent  structure  of  sufficient  capacity  may  subse- 
quently be  constructed  within  the  temporary  structure. 

180.  Empirical  formulae.  Two  of  the  best  known  empirical 
formulae  for  area  of  the  waterway  are  the  following : 

(a)  Myer's  formula: 

Area  of  waterway  in  square  feet  =  C  X  '/drainage  area  in  acres, 
where  C  is  a  coefficient  varying  from  1  for  flat  country  to  4  for 
mountainous  country  and  rocky  ground.  As  an  illustration,  if 
the  drainage  area  is  100  acres,  the  waterway  area  should  be  from 
10  to  40  square  feet,  according  to  the  value  of  the  coefficient 
chosen.  It  should  be  noted  that  this  formula  does  not  regard 
the  great  variations  in  rainfall  in  various  parts  of  the  world  nor 
the  design  of  the  culvert,  and  also  that  the  final  result  depends 
largely  on  the  choice  of  the  coefficient. 

"(b)  Talbot's  formula: 

Area  of  waterway  in  square  feet  =  C  X  '/(drainage  area  in  acres)3. 
44  For  steep  and  rocky  ground  C  varies  from  f  to  1 .  For  rolling 
agricultural  country  subject  to  floods  at  times  of  melting  snow, 
and  with  the  length  of  the  valley  three  or  four  times  its  width,  C 
is  about  \ ;  and  if  the  stream  is  longer  in  proportion  to  the  area, 
decrease  C.  In  districts  not  affected  by  accumulated  snow,  and 


206  RAILROAD   CONSTRUCTION.  §  181. 

where  the  length  of  the  valley  is  several  times  the  width,  £  or  £, 
or  even  less,  may  be  used.  C  should  be,  increased  for  steep  side 
slopes,  especially  if  the  upper  part  of  the  valley  has  a  much 
greater  fall  than  the  channel  at  the  culvert. "  *  As  an  illustration, 
if  the  drainage  area  is  100  acres  the  area  of  waterway  should  be 
C  X  31.6.  The  area  should  then  vary  from  5  to  31  square 
feet,  according  to  the  character  of  the  country.  Like  the 
previous  estimate,  the  result  depends  on  the  choice  of  a  coef- 
ficient and  disregards  local  variations  in  rainfall,  except  as  they 
may  be  arbitrarily  allowed  for  in  choosing  the  coefficient. 

181.  Value  of  empirical  formulae.     The  fact  that  these  for- 
mulae, as  well  as  many  others  of  similar  nature  that  have  been 
suggested,  depend  so  largely  upon  the  choice  of  the  coefficient 
shows  that  they  are  valuable  ' '  more  as  a  guide  to  the  judgment 
than  as  a  working  rule,"  as  Prof.  Talbot  explicitly  declares  in 
commenting  on  his  own  formula.      In  short,  they  are  chiefly  valu- 
able in  indicating  a  probable  maximum  and  minimum  between 
which  the  true  result  probably  lies. 

182,  Results  based  on  Observation,     As  already  indicated  in 
§   179,  observation  of  the  stream  in   question  gives    the  most 
reliable  results.     If  the  country  is  new  and  no  records  of  the 
flow  of  the  stream  during  heavy  storms  has  been  taken,  even 
the  life  of  a  temporary  wooden   structure   may  not   be   long 
enough  to  include  one  of  the  unusually  severe  storms  which 
must  be  allowed  for,  but  there  will  usually  be  some  high-water 
mark  which  will  indicate  how  much  opening  will  be  required. 
The  following  quotation  illustrates  this :    "A  tidal  estuary  may 
generally  be  safely   narrowed   considerably    from   the  extreme 
water    lines    if    stone    revetments    are    used    to    protect   the 
bank  from  wash.     Above  the  true  estuary,  where  the  stream 
cuts  through  the  marsh,  we  generally  find  nearly  vertical  banks, 
and  we  are  safe  if  the  faces  of  abutments  are  placed  even  with 
the  banks.     In  level  sections  of  the  country,  where  the  current 
is  sluggish,    it  is  usually   safe  to    encroach    somewhat   on  the 

*Prof.  A.  N.  Talbot,  "Selected  Papers  of  the  Civil  Engineers'  Club  of 
the  Univ.  of  Illinois." 


§  183.  CULVERTS  AND  MINOR  BRIDGES.  207 

general  width  of  the  stream,  but  in  rapid  streams  among  the 
hills  the  width  that  the  stream  has  cut  for  itself  through  the 
soil  should  not  be  lessened,  and  in  ravines  carrying  mountain 
torrents  the  openings  must  be  left  very  much  larger  than  the 
ordinary  appearance  of  the  banks  of  the  stream  would  seem  to 
make  necessary."  * 

As  an  illustration  of  an  observation  of  a  storm- flow  through 
a  temporary  trestle,  the  following  is  quoted:  "Having  the 
flood  height  and  velocity,  it  is  an  easy  matter  to  determine  the 
volume  of  water  to  be  taken  care  of.  I  have  one  ten- bent  pile 
trestle  135  feet  long  and  24  feet  high  over  a  spring  branch  that 
ordinarily  runs  about  six  cubic  inches  per  second.  Last  sum- 
mer during  one  of  our  heavy  rainstorms  (four  inches  in  less 
than  three  hours)  I  visited  this  place  and  found  by  float  observa- 
tions the  surface  velocity  at  the  highest  stage  to  be  1.9  feet  per 
second.  I  made  a  high- water  mark,  and  after  the  flood- water 
receded  found  the  width  of  stream  to  be  12  feet  and  an  average 
depth  of  2J  feet.  This,  with  a  surface  velocity  of  1.9  feet  per 
second,  would  give  approximately  a  discharge  of  50  cubic  feet, 
or  375  gallons,  per  second.  Having  this  information  it  is  easy 
to  determine  size  of  opening  required."  f 

183.  Degree  of  accuracy  required.  The  advantages  result- 
ing from  the  use  of  standard  designs  for  culverts  (as  well  as 
other  structures)  have  led  to  the  adoption  of  a  comparatively 
small  number  of  designs.  The  practical  use  made  of  a  compu- 
tation of  required  waterway  area  is  to  determine  which  one  of 
several  standard  designs  will  most  nearly  fulfill  the  require- 
ments. For  example,  if  a  24-inch  iron  pipe,  having  an  area  of 
3.14  square  feet,  is  considered  to  be  a  little  small,  the  next  size 
(30-inch)  would  be  adopted ;  but  a  30-inch  pipe  has  an  area  of 
4.92  square  feet,  which  is  56$  larger.  A  similar  result,  except 
that  the  percentage  of  difference  might  not  be  quite  so  marked, 

*  J.  P.  Snow,  Boston  &  Maine  Railway.  From  Report  to  Association  of 
Railway  Superintendents  of  Bridges  and  Buildings.  1897. 

f  A.  J.  Kelley,  Kansas  City  Belt  Railway.  From  Report  to  Association 
of  Railwa)  Superintendents  of  Bridges  and  Buildings.  1897. 


208  RAILED  AD  CONSTRUCTION.  §  184. 

will  be  found  by  comparing  the  areas  of  consecutive  standard 
designs  for  stone  box  culverts. 

The  advisability  of  designing  a  culvert  to  withstand  any 
storm-flow  that  may  ever  occur  is  considered  doubtful.  Several 
years  ago  a  record-breaking  storm  in  New  England  carried 
away  a  very  large  number  of  bridges,  etc.,  hitherto  supposed  to 
be  safe.  It  was  not  afterward  considered  that  the  design  of 
those  bridges  was  faulty,  because  the  extra  cost  of  constructing 
bridges  capable  of  withstanding  such  a  flood,  added  to  interest 
for  a  long  period  of  years,  would  be  enormously  greater  than 
the  cost  of  repairing  the  damages  of  such  a  storm  once  or  twice 
in  a  century.  Of  course  the  element  of  danger  has  some 
weight,  but  not  enough  to  justify  a  great  additional  expendi- 
ture, for  common  prudence  would  prompt  unusual  precautions 
during  or  immediately  after  such  an  extraordinary  storm. 

PIPE    CULVERTS. 

184.  Advantages.      Pipe   culverts,    made    of   cast    iron    or 
earthenware,  are  very  durable,  readily  constructed,  moderately 
cheap,  will  pass  a  larger  volume  of  water  in  proportion  to  the 
area  than  many  other  designs  on  account  of  the  smoothness  of 
the  surface,  and  (when  using  iron  pipe)  may  be  used  very  close 
to  the  track  when  a  low  opening  of  large  capacity  is  required. 
Another  advantage  lies  in  the  ease  with  which  they  may  be  in- 
serted through  a  somewhat  larger  opening  that  has  been  tem- 
porarily lined   with  wood,  without  disturbing  the   roadbed  or 
track. 

185.  Construction.     Permanency  requires  that  the  founda- 
tion shall  be  firm  and  secure  against  being  washed  out.     To 
accomplish  this,  the  soil  of  the  trench  should  be  hollowed  out  to 
fit  the  lower  half  of  the  pipe,  making  suitable  recesses  for  the 
bells."    In  very  soft  treacherous  soil  a  foundation -block  of  con- 
crete is  sometimes  placed  under  each  joint,  or  even  throughout 
the  whole  length.     "When   pipes   are   laid  through  a  slightly 
larger  timber  culvert  great  care  should  be  taken  that  the  pipes 
are  properly  supported,   so  that  there  will  be  no  settling  nor 


§  186.  CULVERTS  AND  MINOR  BRIDGES.  209 

development  of  unusual  strains  when  the  timber  finally  decays 
and  gives  way.  To  prevent  the  washing  away  of  material 
around  the  pipe  the  ends  should  be  protected  by  a  bulkhead. 
This  is  best  constructed  of  masonry  (see  Fig.  97),  although  wood 
is  sometimes  used  for  cheap  and  minor  constructions.  The  joints 
should  be  calked,  especially  when  the  culvert  is  liable  to  run 
full  or  when  the  outflow  is  impeded  and  the  culvert  is  liable  to 
be  partly  or  wholly  filled  during  freezing  weather.  The  cost  of 
a  calking  of  clay  or  even  hydraulic  cement  is  insignificant  com- 
pared with  the  value  of  the  additional  safety  afforded.  When 
the  grade  of  the  pipe  is  perfectly  uniform,  a  very  low  rate  of 
grade  will  suffice  to  drain  a  pipe  culvert,  but  since  some  uneven- 
ness  of  grade  is  inevitable  through  uneven  settlement  or  im- 
perfect construction,  a  grade  of  1  in  20  should  preferably  be 
required,  although  much  less  is  often  used.  The  length  of  a 
pipe  culvert  is  approximately  determined  as  follows : 

Length  =  2s  (depth  of  embankment  to  top  of  pipe)  +  (width  of  roadbed), 

in  which  s  is  the  slope  ratio  (horizontal  to  vertical)  of  the  banks. 
In  practice  an  even  number  of  lengths  will  be  used  which  will 
most  nearly  agree  with  this  formula. 

186.  Iron-pipe  culverts.  Simple  cast-iron  pipes  are  used  in 
sizes  from  12"  to  48"  diameter.  These  are  usually  made  in 
lengths  of  12  feet  with  a  few  lengths  of  6  feet,  so  that  any 
required  length  may  be  more  nearly  obtained.  The  lightest 
pipes  made  are  sufficiently  strong  for  the  purpose,  and  even  those 
which  would  be  rejected  because  of  incapacity  to  withstand  pres- 
sure may  be  utilized  for  this  work.  In  Fig.  97  are  shown  the 
standard  plans  used  on  the  C.  C.  C.  &  St.  L.  E-y.,  which  may 
be  considered  as  typical  plans. 

Pipes  formed  of  cast-iron  segments  have  been  used  up  to  12 
feet  diameter.  The  shell  is  then  made  comparatively  thin,  but 
is  stiffened  by  ribs  and  flanges  on  the  outside.  The  segments 
break  joints  and  are  bolted  together  through  the  flanges.  The 
joints  are  made  tight  by  the  use  of  a  tarred  rope,  together  with 
neat  cement. 


210 


RAILROAD   CONSTRUCTION. 


§186. 


§187. 


CULVERTS  AND  MINOR  BRIDGES. 


211 


187.  Tile-pipe  culverts.  The  pipes  used  for  this  purpose 
vary  from  12"  to  24"  in  diameter.  When  a  larger  capacity 
is  required  two  or  more  pipes  may  be  laid  side  by  side,  but  in 
such  a  case  another  design  might  be  preferable.  It  is  frequently 
specified  that  u  double  -strength  "  or  "  extra-heavy  "  pipe  shall 
be  used,  evidently  with  the  idea  that  the  stresses  on  a  culvert- 
pipe  are  greater  than  on  a  sewer-pipe.  But  it  has  been  con- 
clusively demonstrated  that,  no  matter  how  deep  the  embankment, 
the  pressure  cannot  exceed  a  somewhat  uncertain  maximum, 
also  that  the  greatest  danger  consists  in  placing  the  pipe  so  near 
the  ties  that  shocks  may  be  directly  transferred  to  the  pipe  with- 
out the  cushioning  effect  of  the  earth  and  ballast.  When  the 
pipes  are  well  bedded  in  clear  earth  and  there  is  a  sufficient 
depth  of  earth  over  them  to  avoid  direct  impact  (at  least  three 
feet)  the  ordinary  sewer-pipe  will  be  sufficiently  strong. 
* '  Double-strength  "  pipe  is  frequently  less  perfectly  burned,  and 


UP-STREAMJEND.      DOWN-STREAM  END.         DOWN-STREAM  END.  THREE  PI 
FIG.  98.— STANDARD  VITBLFIED-PIPE  CULVERT.     PLANT  SYSTEM. 


(1891.) 


the  supposed  extra  strength  is  not  therefore  obtained.  In  Fig. 
98  are  shown  the  standard  plans  for  vitrified- pipe  culverts  as  used 
on  the  "Plant  system."  Tile  pipe  is  much  cheaper  than  iron 
pipe,  but  is  made  in  much  shorter  lengths  and  requires  much 
more  work  in  laying  and  especially  to  obtain  a  uniform  grade. 


212 


RAILROAD  CONSTRUCTION. 


§188. 


BOX    CULVEKTS. 

188.  Wooden  box  culverts,  This  form  serves  the  purpose  of 
a  cheap  temporary  construction  which  allows  the  use  of  a  bal- 
lasted roadbed.  As  in  all  temporary  constructions,  the  area 
should  be  made  considerably  larger  than  the  calculated  area 
(§§  179-182),  not  only  for  safety  but  also  in  order  that,  if  the 
smaller  area  is  demonstrated  to  be  sufficiently  large,  the  per- 
manent construction  (probably  pipe)  may  be  placed  inside  with- 
out disturbing  the  embankment.  All  designs  agree  in  using 
heavy  timbers  (12"  X  12",  10"  X  12",  or  8"  X  12")  for  the 
side  walls,  cross-timbers  for  the  roof,  every  fifth  or  sixth  timber 
being  notched  down  so  as  to  take  up  the  thrust  of  the  side  walls, 
and  planks  for  the  flooring.  Fig.  99  shows  some  of  the  standard 
designs  as  used  by  the  C. ,  M.  &  St.  P.  Ry. 


NOTE:-FOR  e  COVERING,  EVERY  SIXTH  STICK  s  THICK. 


FIG.  99.— STANDARD  TIMBER  Box  CULVERT.   C.,  M.  &  ST.  P.  RY.   (Feb.  1889. > 

189.  Stone  box  culverts.  In  localities  where  a  good  quality 
of  stone  is  cheap,  stone  box  culverts  are  the  cheapest  form  of 
permanent  construction  for  culverts  of  medium  capacity,  but 
their  use  is  decreasing  owing  to  the  frequent  difficulty  in  obtain- 
ing really  suitable  stone  within  a  reasonable  distance  of  the 
culvert.  The  clear  span  of  the  cover-stones  varies  from  2  to  4 
feet.  The  required  thickness  of  the  cover-stones  is  sometimes 
calculated  by  the  theory  of  transverse  strains  on  the  basis  of  cer- 
tain assumptions  of  loading — as  a  function  of  the  height  of  the 
embankment  and  the  unit  strength  of  the  stone  used.  Such  a 
method  is  simply  another  illustration  of  a  class  of  calculations 


§  190.  CULVERTS  AND  MINOR  BRIDGES.  213 

which  look  very  precise  and  beautiful,  but  which  are  worse  than 
useless  (because  misleading)  on  account  of  the  hopeless  uncertainty 
as  to  the  true  value  of  certain  quantities  which  must  be  used  in 
the  computations.  In  the  first  place  the  true  value  of  the  unit 
tensile  strength  of  stone  is  such  an  uncertain  and  variable 
quantity  that  calculations  based  on  any  assumed  value  for  it  are 
of  small  reliability.  In  the  second  place  the  weight  of  the  prism 
of  earth  lying  directly  above  the  stone,  plus  an  allowance  for  live 
load,  is  by  no  means  a  measure  of  the  load  on  the  stone  nor  of 
the  forces  that  tend  to  fracture  it.  All  earthwork  will  tend  to 
form  an  arch  above  any  cavity  and  thus  relieve  an  uncertain  and 
probably  variable  proportion  of  the  pressure  that  might  other- 
wise exist.  The  higher  the  embankment  the  less  the  propor- 
tionate loading,  until  at  some  uncertain  height  an  increase  in 
height  will  not  increase  the  load  on  the  cover-stones.  The  effect 
of  frost  is  likewise  large,  but  uncertain  and  not  computable.  The 
usual  practice  is  therefore  to  make  the  thickness  such  as  experi- 
ence has  shown  to  be  safe  with  a  good  quality  of  stone,  i.e., 
about  10  or  12  inches  for  2  feet  span  and  up  to  16  or  18  inches 
for  4  feet  span.  The  side  walls  should  be  carried  down  deep 
enough  to  prevent  their  being  undermined  by  scour  or  heaved 
by  frost.  The  use  of  cement  mortar  is  also  an  important  feature 
of  first-class  work,  especially  when  there  is  a  rapid  scouring  cur- 
rent or  a  liability  that  the  culvert  will  run  under  a  head.  In 
Fig.  100  are  shown  standard  plans  for  single  and  double  stone  box 
culverts  as  used  on  the  Norfolk  and  Western  R.R. 

190.  Old-rail  culverts.  It  sometimes  happens  (although  very 
rarely)  that  it  is  necessary  to  bring  the  grade  line  within  3  or  4 
feet  of  the  bottom  of  a  stream  and  yet  allow  an  area  of  10  or  12 
square  feet.  A  single  large  pipe  of  sufficient  area  could  not  be 
used  in  this  case.  The  use  of  several  smaller  pipes  side  by  side 
would  be  both  expensive  and  inefficient.  For  similar  reasons 
neither  wooden  nor  stone  box  culverts  could  be  used.  In  such 
cases,  as  well  as  in  many  others  where  the  head-room  is  not  so 
limited,  the  plan  illustrated  in  Fig.  101  is  a  very  satisfactory 
solution  of  the  problem.  The  old  rails,  having  a  length  of  8  or 


214 


RAILROAD   CONSTRUCTION. 


§190. 


•I 

a 

I 

< 

p 
E 

I 

^    r-i 


191. 


CULVERTS  AND  MINOR  BRIDGES. 


215 


9  feet,  are  laid  close  together  across  a  6 -foot  opening.  Some- 
times the  rails  are  held  together  by  long  bolts  passing  through 
the  webs  of  the  rails.  In  the  plan  shown  the  rails  are  confined 


FIG.  101.— STANDARD  OLD-RAIL  CULVERT.     N.  &  W.  R.R.     (1895.) 

by  low  end  walls  on  each  abutment.  This  plan  requires  only  1 5 
inches  between  the  base  of  the  rail  and  the  top  of  the  culvert 
channel.  It  also  gives  a  continuous  ballasted  roadbed. 


AKCH    CULVERTS. 


191.  Influence  of  design  on  flow.  The  variations  in  the 
design  of  arch  culverts  have  a  very  marked  influence  on  the 
cost  and  efficiency.  To  combine  the  least  cost  with  the  great- 
est efficiency,  due  weight  should  be  given  to  the  following 
elements:  (a)  the  amount  of  masonry,  (5)  the  simplicity  of 
the  constructive  work,  (c)  the  design  of  the  wing  walls,  (d) 
the  design  of  the  junction  of  the  wing  walls  with  the  barrel 


(c) 


FIG.  102. — TYPES  OF  CULVERTS. 


and  faces  of  the  arch,  and  (e)  the  safety  and  permanency  of  the 
construction.  These  elements  are  more  or  less  antagonistic  to 
each  other,  and  the  defects  of  most  designs  are  due  to  a  lack  of 
proper  proportion  in  the  design  of  these  opposing  interests.  The 
simplest  construction  (satisfying  elements  b  and  e)  is  the  straight 


216  RAILROAD   CONSTRUCTION.  §  192. 

barrel  arch  between  two  parallel  vertical  head  walls,  as  sketched 
in  Fig.  102,  a.  From  a  hydraulic  standpoint  the  design  is  poor, 
as  the  water  eddies  around  the  corners,  causing  a  great  resistance 
which  decreases  the  flow.  Fig.  102,  J,  shows  a  much  better  de- 
sign in  many  respects,  but  much  depends  on  the  details  of  the 
design  as  indicated  in  elements  (b)  and  (d).  As  a  general  thing 
a  good  hydraulic  design  requires  complicated  and  expensive 
masonry  construction,  i.e.,  elements  (fj)  and  (d)  are  opposed. 
Design  102,  <?,  is  sometimes  inapplicable  because  the  water  is 
liable  to  work  in  behind  the  masonry  during  floods  and  perhaps 
cause  scour.  This  design  uses  less  masonry  than  (a)  or  (5). 

192.  Example  of  arch  culvert  design.     In  Plate  XY  is  shown 
the  design  for  an  8 -foot  arch  culvert  according  to  the  standard 
of  the    Norfolk   and  Western  R.R.     Note  that  the  plan  uses 
the   flaring  wing  walls    (Fig.   102,   b)    on    the   up-stream    side 
(thus  protecting  the  abutments  from  scour)  and  straight  wing 
walls  (similar  to  Fig.  102,  c)  on  the  down-stream  end.     This 
economizes  masonry  and  also  simplifies  the  constructive  work. 
Note  also  the  simplicity  of  the  junction  of  the  wing  walls  with 
the  barrel  of  the  arch,  there  being  no  re-entrant  angles  below 
the  springing  line  of  the  arch.     The  design  here  shown  is  but 
one  of  a  set  of  designs  for  arches  varying  in  span  from  6'  to  30'. 

MINOK    OPENINGS. 

193.  Cattle-guards,     (a)  Pit   guards.     Cattle-guards  will  be 
considered   under  the  head  of  minor  openings,  since  the  old- 
fashioned  plan  of  pit  guards,  which  are  even  now  defended  and 
preferred  by  some  railroad   men,  requires  a  break  in  the  con- 
tinuity of  the  roadbed.     A  pit  about  three  feet  deep,  five  feet 
long,  and  as  wide  as  the  width  of  the  roadbed,  is  walled  up  with 
stone  (sometimes  with  wood),  and  the  rails  are  supported  on  heavy 
timbers  laid   longitudinally  with   the  rails.     The   break  in  the 
continuity  of  the  roadbed  produces  a  disturbance  in  the  elastic 
wave  running  through  the  rails,  the  effect  of  which  is  noticeable 
at  high  velocities.     The  greatest  objection,  however,  lies  in  the 


M 


§193. 


CULVERTS  ^AND  MINOR  BRIDGES. 


217 


dangerous  consequences  of  a  derailment  or  a  failure  of  the  tim- 
bers owing  to  unobserved  decay  or  destruction  by  fire — caused 
perhaps  by  sparks  and  cinders  from  passing  locomotives.  The 
very  insignificance  of  the  structure  often  leads  to  careless  in- 


•4*xl3*x7'o" 


FIG.  103.— PIT  CATTLE-GUARDS.    P.  R.R. 

spection.  But  if  a  single  pair  of  wheels  gets  off  the  rails  and 
drops  into  the  pit,  a  costly  wreck  is  inevitable.  The  (once) 
standard  design  for  such  a  structure  on  the  Pennsylvania  K.R. 
is  shown  in  Fig.  103. 

(b)  Surface  cattle -guards.  These  are  fastened  on  top  of  the 
ties;  the  continuity  of  the  roadbed  is  absolutely  unbroken  and 
thus  is  avoided  much  of  the  danger  of  a  bad  wreck  owing  to  a 
possible  derailment.  The  device  consists  essentially  of  overlay- 
ing the  ties  (both  inside  and  outside  the  rails)  with  a  surface  on 
which  cattle  will  not  walk.  The  multitudinous  designs  for  such 
a  surface  are  variously  effective  in  this  respect.  An  objection, 
which  is  often  urged  indiscriminately  against  all  such  designs,  is 
the  liability  that  a  brake- chain  which  may  happen  to  be  drag- 
ging may  catch  in  the  rough  bars  which  are  used.  The  bars 
are  sometimes  "  home-made,"  of  wood,  as  shown  in  Fig.  104. 
Iron  or  steel  bars  are  made  as  shown  in  Fig.  105.  The 
general  construction  is  the  same  as  for  the  wooden  bars.  The 


218 


RAILROAD   CONSTRUCTION. 


§194. 


metal  bars  have  far  greater  durability,  and  it  is  claimed  that  they 
are  more  effective  in  discouraging  cattle  from  attempting  to 
cross. 


FIG.  104.— CATTLE-GUARD  WITH  WOODEN  SLATS. 

r~\ 


FIG.  105. — MERRILL- STEVENS  STEEL  CATTLE-GUARD. 

194.  Cattle-passes.  Frequently  when  a  railroad  crosses  a 
farm  on  an  embankment,  cutting  the  farm  into  two  parts,  the 
railroad  company  is  obliged  to  agree  to  make  a  passageway 
through  the  embankment  sufficient  for  the  passage  of  cattle  and 
perhaps  even  farm-wagons.  If  the  embankment  is  high  enough 
so  that  a  stone  arch  is  practicable,  the  initial  cost  is  the  only 
great  objection  to  such  a  construction ;  but  if  an  open  wooden 
structure  is  necessary,  all  the  objections  against  the  old-fashioned 
cattle-guards  apply  with  equal  force  here.  The  avoidance  of  a 
grade  crossing  which  would  otherwise  be  necessary  is  one  of  the 


PLATE   XVI. 


(%  BOLT  E'/ERY  THIRD  TIE. 


15  I-BEAM   (DOUBLE,)   30  LBS.   PER  FOOT 
17  FT.   LONG,   14  FEET  CLEAR  SPAN. 


STANDARD  I-BRIDGES— 14-FT.  SPAN. 

NORFOLK  AND  WESTERN  R.R. 
(1891.) 


TYPES  OF  PLATE  GIRDER  BRIDGES 


(To  face  page  219.) 


§  195.  CULVERTS^  AND  MINOR  BRIDGES.  219 

great  compensations  for  the  expense  of  the  construction  and 
maintenance  of  these  structures.  The  construction  is  sometimes 
made  by  placing  two  pile  trestle  bents  about  6  to  8  feet  apart, 
supporting  the  rails  by  stringers  in  the  usual  way,  the  special 
feature  of  this  construction  being  that  the  embankments  are 
filled  in  behind  the  trestle  bents,  and  the  thrust  of  the  embank- 
ments is  mutually  taken  up  through  the  stringers,  which  are 
notched  at  the  ends  or  otherwise  constructed  so  that  they  may 
take  up  such  a  thrust.  The  designs  for  old-rail  culverts  and 
arch  culverts  are  also  utilized  for  cattle-passes  when  suitable  and 
convenient,  as  well  as ,  the  designs  illustrated  in  the  following 
section. 

195.  Standard  stringer  and  I-beam  bridges.  The  advantages 
of  standard  designs  apply  even  to  the  covering  of  short  spans 
with  wooden  stringers  or  with  I  beams — especially  since 
the  methods  do  not  require  much  vertical  space  between  the 
rails  and  the  upper  side  of  the  clear  opening,  a  feature  which  is 
often  of  prime  importance.  These  designs  are  chiefly  used  for 
culverts  or  cattle-passes  and  for  crossing  over  highways — pro- 
viding such  a  narrow  opening  would  be  tolerated.  The  plans 
all  imply  stone  abutments,  or  at  least  abutments  of  sufficient 
stability  to  withstand  all  thrust  of  the  embankments.  Some  of 
the  designs  are  illustrated  in  Plate  XYI.  The  preparation  of 
these  standard  designs  should  be  attacked  by  the  same  general 
methods  as  already  illustrated  in  §  156.  When  computing  the 
required  transverse  strength,  due  allowance  should  be  made  for 
lateral  bracing,  which  should  be  amply  provided  for.  Note 
particularly  the  methods  of  bracing  illustrated  in  Plate  XYI. 
The  designs  calling  for  iron  (or  steel)  stringers  may  be  classed 
as  permanent  constructions,  which  are  cheap,  safe,  easily  in- 
spected and  maintained  and  therefore  a  desirable  method  of 
construction. 


CHAPTER  VII. 

BALLAST. 

196.  Purpose  and  requirements,     "  The  object  of  the  ballast 
is  to  transfer  the  applied  load  over  a  large  surface ;   to  hold  the 
timber  work  in  place  horizontally ;   to  carry  oft'  the  rain-water 
from  the  superstructure  and  to  prevent  freezing  up  in  winter ; 
to  afford  means  of  keeping  the  ties  truly  up  to  the  grade  line ; 
and  to  give  elasticity  to  the  roadbed."     This  extremely  con- 
densed statement  is  a  description  of  an  ideally  perfect  ballast. 
The  value  of  any  given  kind  of  ballast  is  proportional  to  the 
extent  to  which  it  fulfills  these  requirements.     The  ideally  per- 
fect ballast  is  not  necessarily  the  most  economical  ballast  for  all 
roads.     Light  traffic  generally  justifies  something  cheaper,  but 
a  very  common  error  is  to  use  a  very  cheap  ballast  when  a  small 
additional  expenditure  would    procure    a   much   better   ballast 
which  would  be  much  more  economical  in  the  long  run. 

197.  Materials.    The  materials  most  commonly. employed  are 
gravel  and  broken  stone.     Burnt  clay,  cinders,  shells,  and  small 
coal  are  occasionally  used  as  ballast  when  they  are  especially 
cheap  and  convenient  or  when  better  kinds  are  especially  expen- 
sive.    Although  it  is  hardly  correct  to  speak  of  the  natural  soil 
as  ballast,  yet  many  miles  of  cheap  railways  are  ' '  ballasted 
with  the  natural  soil,  which  is  then  called  "  mud  ballast. 

Mud  ballast.  When  the  natural  soil  is  gravelly  so  that  rain 
will  drain  through  it  quickly,  it  will  make  a  fair  roadbed  for 
light  traffic,  but  for  heavy  traffic,  and  for  the  greater  part  of  the 
length  of  most  roads,  the  natural  soil  is  a  very  poor  material  for 
ballast ;  for,  no  matter  how  suitable  the  soil  might  be  along 

220 


55 
'l/CU. 

5  5 


§  197.  BALLAST.  221 

limited  sections  of  the  road,  it  would  practically  never  happen 
that  the  soil  would  be  uniformly  good  throughout  the  whole 
length  of  the  road.  Considering  that  a  heavy  rain  will  in  one 
day  spoil  the  results  of  weeks  of  patient  ' '  surfacing ' '  with  mud 
ballast,  it  is  seldom  economical  to  use  "mud"  if  there  is  a 
gravel-bed  or  other  source  of  ballast  anywhere  on  the  line  of 
the  road. 

Cinders.  The  advantages  consist  in  the  excellent  facilities 
for  drainage,  ease  of  handling,  and  cheapness — after  the  road  is 
in  operation.  One  disadvantage  is  excessive  dust  in  dry  weather. 
Cinders  are  considered  preferable  to  gravel  in  yards. 

Slag.  "When  slag  is  readily  obtainable  it  furnishes  an  ex- 
cellent ballast,  free  from  dust  and  perfect  in  drainage  qualities. 
Some  kinds  of  slag  are  objectionable  on  account  of  their  delete- 
rious chemical  effect  on  the  ties  and  spikes — especially  on 
metallic  ties. 

Shells,  small  coal,  etc.  These  comparatively  inferior  kinds 
of  ballast  are  used  for  light  traffic  when  they  are  especially  cheap 
and  convenient.  They  are  extremely  dusty  in  dry  weather, 
break  up  into  very  fine  dust,  and  are  but  little  better  than  mud. 

Gravel.  This  is  the  most  common  form  of  ballast  which 
may  be  called  good  ballast.  In  1885,  the  Roadmasters  Associa- 
tion of  America  voted  in  favor  of  gravel  ballast  as  against  rock 
ballast.  Although  not  so  stated,  this  action  was  perhaps  due  to 
a  conviction  of  its  real  economy  for  the  average  railroad  of  this 
country,  which  may  be  called  a  "light  traffic"  road.  Gravel 
should  preferably  be  screened  over  a  screen  having  a  J"  mesh, 
so  as  to  screen  out  all  dirt  and  the  finest  stones.  Generally  a 
railroad  will  be  able  to  find  at  some  point  along  its  line  a 
"gravel-pit"  affording  a  suitable  supply.  This  may  be  dug  out 
with  a  steam-shovel,  screened  if  necessary,  and  sent  out  over 
the  line  by  the  train-load  at  a  comparatively  small  cost. 

Rock  or  broken  stone.  Eock  ballast  is  generally  specified  to 
be  such  as  will  pass  through  a  1 J"  (or  2")  ring.  Although  pref- 
erably broken  by  hand,  machine-broken  stone  is  much  cheaper. 
It  is  most  easily  handled  with  forks.  This  also  has  the  effect  of 


222  RAILROAD  CONSTRUCTION.  §  198. 

screening  out  the  dirt  and  fine  chips  which  would  interfere  with 
effectual  drainage.  Rock  ballast  is  more  expensive  in  first  cost, 
and  also  more  troublesome  to  handle,  than  any  other  kind,  but 
under  heavy  traffic  will  keep  in  surface  better  and  will  require 
less  work  for  maintenance  after  the  ties  have  become  thoroughly 
bedded.  For  roads  with  very  light  traffic,  running  few  trains, 
at  comparatively  low  velocities,  the  advantages  of  rock  ballast 
over  other  kinds  are  not  so  pronounced.  For  such  roads  rock 
ballast  is  an  expensive  luxury.  The  amount  of  traffic  which 
will  justify  the  use  of  rock  ballast  will  depend  on  the  cost  of 
obtaining  ballast  of  the  various  kinds. 

198.  Cross-sections.  A  depth  of  12"  under  the  tie  is  gener- 
ally required  on  the  best  roads,  but  for  light  traffic  this  is  some- 
times reduced  to  6"  and  even  less.  The  width  is  generally  1  to 
2  feet  less  than  the  width  of  the  roadbed  proper — excluding 
ditches.  If  the  ballast  has  an  average  width  of  10  feet  (12  feet 
at  bottom  and  8  feet  at  top)  and  an  average  depth  of  15  inches 
(including  that  placed  between  the  ties),  it  will  require  2444 
cubic  yards  per  mile  of  track.  The  P.  R.R.  estimates  2500 
cubic  yards  of  gravel  and  2800  cubic  yards  of  stone  ballast  per 
mile  of  single  track.  On  account  of  the  requirements  of  drain- 
age the  best  form  of  cross-section  depends  on  the  kind  of  ballast 
used. 

Mud  ballast.  Since  the  great  objection  to  mud  ballast  lies  in 
its  liability  to  become  soft  by  soaking  up  the  rain  that  falls,  it 
becomes  necessary  that  it  should  be  drained  as  quickly  and 
readily  as  its  nature  will  permit.  Fig.  106  shows  a  typical 


FIG.  ..06.—"  MUD  "  BALLAST. 


cross-section  for  mud  ballast.  It  should  be  crowned  2"  above 
the  top  of  the  tie  at  the  center,  thence  sloped  so  as  to  leave  a 
slight  clearance  under  the  rail  between  the  ties,  thence  sloping 
down  to  the  bottom  of  the  tie  at  each  end  and  continuing  to 


§199. 


BALLAST. 


223 


slope  down  to  the  ditch  (in  cut),  which  should  be  18"  or  20"  be- 
low the  bottom  of  the  tie. 

Gravel,  cinders,  slag,  etc.     The  subgrade  is  crowned  6"  or 
8"  in  the  center,  as  shown  in  Fig.  107.     The  ballast  is  crowned 


FIG.  107.— GRAVEL  BALLAST. 

to  the  top  of  the  tie  in  the  center,  but  is  sloped  down  to  the 
bottom  of  the  tie  at  each  end.  This  is  necessary  (and  more 
especially  so  with  mud  ballast)  to  prevent  a  possible  accumula- 
tion and  settlement  of  water  at  the  ends  of  the  tie,  which  would 
readily  soak  into  the  end  fibers  and  produce  decay. 

Broken  stone.  Stone  ballast  is  shouldered  out  beyond  the 
ends  of  the  ties  so  as  to  afford  greater  lateral  binding.  The 
space  between  the  ties  is  filled  up  level  with  the  tops.  The 


FIG.  108. — BROKEN  STONE  BALLAST. 

perfect  drainage  of  stone  ballast  permits  this  to  be  done  without 
any  danger  of  causing  decay  of  the  ties  by  the  accumulation  and 
retention  of  water. 

199,  Methods  of  laying  ballast.  The  cheapest  method  of 
laying  ballast  on  new  roads  is  to  lay  ties  and  rails  directly  on 
the  prepared  subgrade  and  run  a  construction  train  over  the 
track  to  distribute  the  ballast.  Then  the  track  is  lifted  up  until 
sufficient  ballast  is  worked  under  the  ties  and  the  track  is  prop- 
erly surfaced.  This  method,  although  cheap,  is  apt  to  injure 
the  rails  by  causing  bends  and  kinks,  due  to  the  passage  of 
loaded  construction  trains  when  the  ties  are  very  unevenly  and 
roughly  supported,  and  the  method  is  therefore  condemned  and 
prohibited  in  some  specifications.  The  best  method  is  to  draw 


224  RAILROAD  CONSTRUCTION.  §  200. 

in  carts  (or  on  a  contractor's  temporary  track)  the  ballast  that  is 
required  under  the  level  of  the  bottom  of  the  ties.  Spread  this 
ballast  carefully  to  the  required  surface.  Then  lay  the  ties  and 
rails,  which  will  then  have  a  very  fair  surface  and  uniform  sup- 
port. A  construction  train  can  then  be  run  on  the  rails  and 
distribute  sufficient  additional  ballast  to  pack  around  and  between 
the  ties  and  make  the  required  cross-section. 

The  necessity  for  constructing  some  lines  at  an  absolute 
minimum  of  cost  and  of  opening  them  for  traffic  as  soon  as  pos- 
sible has  often  led  to  the  policy  of  starting  traffic  when  there  is 
little  or  no  ballast — perhaps  nothing  more  than  a  mere  tamping 
of  the  natural  soil  under  the  ties.  When  this  is  done  ballast 
may  subsequently  be  drawn  where  required  by  the  train-load  on 
flat  cars  and  unloaded  at  a  minimum  of  cost  by  means  of  a 
"  plough."  The  plough  has  the  same  width  as  the  cars  and  is 
guided  either  by  a  ridge  along  the  center  of  each  car  or  by  short 
posts  set  up  at  the  sides  of  the  cars.  It  is  drawn  from  one  end 
of  the  train  to  the  other  by  means  of  a  cable.  The  cable  is 
sometimes  operated  by  means  of  a  small  hoisting-engine  carried 
on  a  car  at  one  end  of  the  train.  Sometimes  the  locomotive  is 
detached  temporarily  from  the  train  and  is  run  ahead  with  the 
cable  attached  to  it. 

200.  Cost.  The  cost  of  ballast  in  the  track  is  quite  a  variable 
item  for  different  roads,  since  it  depends  (a)  on  the  first  cost  of 
the  material  as  it  comes  to  the  road,  (£>)  on  the  distance  from 
the  source  of  supply  to  the  place  where  it  is  used,  and  (c)  on 
the  method  of  handling.  The  first  cost  of  cinder  or  slag  is 
frequently  insignificant.  A  gravel-pit  may  cost  nothing  except 
the  price  of  a  little  additional  land  beyond  the  usual  limits  of  the 
right  of  way.  Broken  stone  will  usually  cost  $1  or  more  per 
cubic  yard.  If  suitable  stone  is  obtainable  on  the  company's 
land,  the  cost  of  blasting  and  breaking  should  be  somewhat  less 
than  this.  The  cost  of  loading  the  ballast  on  to  trains  will  be 
small  (per  cubic  yard)  if  it  is  handled  with  steam-shovels — as  in 
the  case  of  gravel  taken  from  a  gravel-pit.  Hand-shovelling 
will  cost  more.  The  cost  of  hauling  will  depend  on  the  distance 


§  200  BALLAST. 

hauled,  and  also,  to  a  considerable  extent,  on  the  limitations  on 
the  operation  of  the  train  due  to  the  necessity  of  keeping  out  of 
the  way  of  regular  trains.  There  is  often  a  needless  waste  in 
this  way.  The  "  mud  train  "  is  considered  a  pariah  and  entitled 
to  no  rights  whatever,  regardless  of  the  large  daily  cost  of  such 
a  train  and  of  the  necessary  gang  of  men.  The  cost  of  broken 
stone  ballast  in  the  track  is  estimated  at  $1.25  per  cubic  yard. 
The  cost  of  gravel  ballast  is  estimated  at  60  c.  per  cubic  yard 
in  the  track.  The  cost  of  placing  and  tamping  gravel  ballast  is 
estimated  at  20  c.  to  24  c.  per  cubic  yard,  for  cinders  12  c.  to 
15  c.  per  cubic  yard.  The  cost  of  loading  gravel  on  cars,  using 
a  steam-shovel,  is  estimated  at  6  c.  to  10  c.  per  cubic  yard.* 

*  Report  Headmasters  Association,  1885. 


CHAPTEE  VIII. 

TIES, 

AND  OTHER  FORMS  OF  RAIL  SUPPORT. 

201.  Various  methods  of  supporting  rails.     It   is   necessary 
that  the  rails  shall  be  sufficiently  supported  and  braced,  so  that 
the  gauge  shall  be  kept  constant  and  that  the  rails  shall  not  be 
subjected  to    excessive  transverse  stress.     It  is  also  preferable 
that  the  rail  support  shall  be  neither  rigid  (as  if  on  solid  rock) 
nor  too  yielding,  but  shall  have  a  uniform  elasticity  throughout. 
These  requirements  are  more  or  less  fulfilled  by  the  following 
methods. 

(a)  Longitudinals.     Supporting   the    rails    throughout   their 
entire  length.     This  method  is  very  seldom  used  in  this  country 
except   occasionally    on    bridges    and    in    terminals    when    the 
longitudinals  are   supported   on    cross- ties.       In  §  224  wrill  be 
described  a  system  of  rails,    used  to   some  extent  in  Europe, 
having  such  broad   bases  that  they  are  self-supporting  on  the 
ballast    and    are    only   connected   by  tie-rods    to   maintain  the 
gauge. 

(b)  Cast-iron  "bowls"  or  "pots."     These  are  castings  resem- 
bling large  inverted  bowls  or  pots,   having  suitable  chairs  on 
top  for  holding  and  supporting  the  rails,    and   tied   together 
with  tie-rods.     They  will  be  described  more  fully  later  (§  223). 

(b)  Cross-ties  of  metal  or  wood.     These  will  be  discussed  in 
the  following  sections. 

202.  Economics  of  ties.     The  true  cost  of  ties  depends  on  the 
relative  total  cost  of  maintenance  for  long  periods  of  time.     The 
first  cost  of  the  ties  delivered  to  the  road  is  but  one  item  in  the 

226 


§203.  v      TIES.  227 

economics  of  the  question.  Cheap  ties  require  frequent  renew- 
als, which  cost  for  the  labor  of  each  renewal  practically  the 
same  whether  the  tie  is  of  oak  or  hemlock.  Cheap  ties  make  a 
poor  roadbed  which  will  require  more  track  labor  to  keep  even 
in  tolerable  condition.  The  roadbed  will  require  to  be  disturbed 
so  frequently  on  account  of  renewals  that  the  ties  never  get  an 
opportunity  to  get  settled  and  to  form  a  smooth  roadbed  for  any 
length  of  time.  Irregularity  in  width,  thickness,  or  length  of 
ties  is  especially  detrimental  in  causing  the  ballast  to  act 
and  wear  unevenly.  The  life  of  ties  has  thus  a  more  or  less 
direct  influence  on  the  life  of  the  rails,  on  the  wear  of  rolling 
stock,  and  on  the  speed  of  trains.  These  last  items  are  not  so 
readily  reducible  to  dollars  and  cents,  but  when  it  can  be  shown 
that  the  total  cost,  for  a  long  period  of  time,  of  several  renewals 
of  cheap  ties,  with  all  the  extra  track  labor  involved,  is  as  great  as 
or  greater  than  that  of  a  few  renewals  of  durable  ties,  then  there 
is  no  question  as  to  the  real  economy.  In  the  following  dis- 
cussions of  the  merits  of  untreated  ties  (either  cheap  or  costly), 
chemically  treated  ties,  or  metal  ties",  the  true  question  is  there- 
fore of  the  ultimate  cost  of  maintaining  any  particular  kind  of 
ties  for  an  indefinite  period,  the  cost  including  the  first  cost  of 
the  ties,  the  labor  of  placing  them  and  maintaining  them  to 
surface,  and  the  somewhat  uncertain  (but  not  therefore  non- 
existent) effect  of  frequent  renewals  on  repairs  of  rolling  stock, 
on  possible  speed,  etc. 

WOODEN    TIES. 

203.  Choice  of  wood.  This  naturally  depends,  for  any  partic- 
ular section  of  country,  on  the  supply  of  wood  which  is  most 
readily  available.  The  woods  most  commonly  used,  especially 
in  this  country,  are  oak  and  pine,  oak  being  the  most  durable 
and  generally  the  most  expensive.  Kedwood  is  used  very  ex- 
tensively in  California  and  proves  to  be  extremely  durable,  so 
far  as  decay  is  concerned,  but  it  is  very  soft  and  is  much  injured 
by  ' '  rail-cutting. ' '  This  defect  is  being  partly  remedied  by  the 


228  RAILROAD  CONSTRUCTION.  §  204. 

use  of  tie-plates,  as  will  be  explained  later.  Cedar,  chestnut, 
hemlock,  and  tamarack  are  frequently  used  in  this  country.  In 
tropical  countries  very  durable  ties  are  frequently  obtained  from 
the  hard  woods  peculiar  to  those  countries.  According  to  a  re- 
cent bulletin  of  the  U.  S.  Department  of  Agriculture  the  pro- 
portions of  the  various  kinds  used  in  the  United  States  are  about 
as  follows : 


Oak 60? 

Pine 20 

Cedar. .  6 


Chestnut 5 

Hemlock  and  Tama- 
rack  3 

Redwood 3 


Cypress 2% 

Various 1 

Total..  .   100$ 


204,  Durability.  The  durability  of  ties  depends  on-  the  cli- 
mate; the  drainage  of  the  ballast  ;  the  volume,  weight,  and 
speed  of  the  traffic ;  the  curvature,  if  any ;  the  use  of  tie-plates ; 
the  time  of  year  of  cutting  the  timber ;  the  age  of  the  timber 
and  the  degree  of  its  seasoning  before  placing  in  the  track ;  the 
nature  of  the  soil  in  which  the  timber  was  grown;  and,  chiefly, 
on  the  species  of  wood  employed.  The  variability  in  these 
items  will  account  for  the  discrepancies  in  the  reports  on  the  life 
of  various  woods  used  for  ties. 

White  oak  is  credited  with  a  life  of  5  to  12  years,  depending 
principally  on  the  traffic.  Is  is  both  hard  and  durable,  the 
hardness  enabling  it  to  withstand  the  cutting  tendency  of  the 
rail-flanges,  and  the  durability  enabling  it  to  resist  decay.  Pine 
and  redwood  resist  decay  very  well,  but  are  so  soft  that  they  are 
badly  cut  by  the  rail-flanges  and  do  not  hold  the  spikes  very 
well,  necessitating  frequent  respiking.  Since  the  spikes  must 
be  driven  within  certain  very  limited  areas  on  the  face  of  each 
tie,  it  does  not  require  many  spike-holes  to  u spike-kill"  the 
tie.  On  sharp  curves,  especially  with  heavy  traffic,  the  wheel- 
flange  pressure  produces  a  side  pressure  on  the  rail  tending  to 
overturn  it,  which  tendency  is  resisted  by  the  spike,  aided  some- 
times by  rail-braces.  Whenever  the  pressure  becomes  too  great 
the  spike  will  yield  somewhat  and  will  be  slightly  withdrawn. 
The  resistance  is  then  somewhat  less  and  the  spike  is  soon  so  loose 
that  it  must  be  redriven  in  a  new  hole.  If  this  occurs  very 


§  206.  TIES.  229 

often,  the  tie  may  need  to  be  replaced  long  before  any  decay  has 
set  in.  "When  the  traffic  is  very  light,  the  wood  very  durable, 
and  the  climate  favorable  ties  have  been  known  to  last  25  years. 

205.  Dimensions.     The  usual  dimensions  for  the  best  roads 
(standard  gauge)  are  8'  to  8'  6"  long,  6"  to  7"  thick,  and  8"  to- 
10"  wide  on  top  and  bottom  (if  they  are  hewed)  or  8"  to  9" 
wide  if  they  are  sawed.     For  cheap  roads  and  light  traffic  the 
length  is  shortened  sometimes  to  7'  and  the  cross-section  also  re- 
duced.    On  the  other  hand  a  very  few  roads  use  ties  9'  long. 

Two  objections  are  urged  against  sawed  ties :  first,  that  the 
grain  is  torn  by  the  saw,  leaving  a  woolly  surface  which  induces 
decay ;  and  secondly,  that,  since  timber  is  not  perfectly  straight- 
grained,  some  of  the  fibers  are  cut  obliquely,  exposing  their  ends, 
which  are  thus  liable  to  decay.  The  use  of  a  ' '  planer-saw  ' '  ob- 
viates the  first  difficulty.  Chemical  treatment  of  ties  obviates 
both  of  these  difficulties.  Sawed,  ties  are  more  convenient  to 
handle,  are  a  necessity  on  bridges  and  trestles,  and  it  is  even. 
claimed,  although  against  commonly  received  opinion,  that 
actual  trial  has  demonstrated  that  they  are  more  durable  than 
hewed  ties. 

206.  Spacing.     The  spacing  is  usually  14  to  16  ties  to  a  30- 
foot  rail.     This  number  is  sometimes  reduced  to  12  and  even 
10,  and  on  the  other  hand  occasionally  increased  to  18  or  20  by 
employing  narrower  ties.     There  is  no  economy  in  reducing  the 
number  of  ties  very  much,  since  for  any  required  stiffness  of 
track  it  is  more  economical  to  increase  the  number  of  supports  than 
to  increase  the  weight  of  the  rail.     The  decreasing  cost  of  rails 
and  the  increasing  cost  of  ties  have  materially  changed  the  rela- 
tion between  number  of  ties  and   weight  of  rail  to  produce  a 
given  stiffness  at  minimum  cost,  but  many  roads  have  found  it 
economical  to  employ  a  large  number  of  ties  rather  than  increase 
the  weight  of  the  rail.     On  the  other  hand  there  is  a  practical 
limit  to  the  number  that  may  be  employed,  on  account  of  the 
necessary  space  between  the   ties  that  is  required   for  proper 
tamping.     This  width  is  ordinarily  about  twice  the  width  of  the 
tie.     At  this  rate,  with  light  ties  6"  wide  and  with  12"  clear 


230  RAILROAD  CONSTRUCTION.  §  207. 

space,  there  would  be  20  ties  per  30-foot  rail,  or  3520  per  mile. 
The  smaller  ties  can  generally  be  bought  much  cheaper  (propor- 
tionately) than  the  larger  sizes,  and  hence  the  economy. 

Track  instructions  to  foremen  generally  require  that  the 
spacing  of  ties  shall  not  be  uniform  along  the  length  of  any 
rail.  Since  the  joint  is  generally  the  weakest  part  of  the  rail 
structure,  the  joint  requires  more  support  than  the  center  of  the 
rail.  Therefore  the  ties  are  placed  with  but  8"  or  10"  clear 
space  between  them  at  the  joints,  this  applying  to  3  or  4  ties  at 
each  joint ;  the  remaining  ties,  required  for  each  rail  length,  are 
equally  spaced  along  the  remaining  distance. 

207.  Specifications.  The  specifications  for  ties  are  apt  to 
include  the  items  of  size,  kind  of  wood,  and  method  of  con- 
struction, besides  other  minor  directions  about  time  of  cutting, 
seasoning,  delivery,  quality  of  timber,  etc. 

(a)  Size.     The  particular  size  or  sizes  required  will  be  some- 
what as  indicated  in  §  205. 

(b)  Kind  of  wood.    When  the  kind  or  kinds  of  wood  are  spe- 
cified, the  most  suitable  kinds  that  are  available  in  that  section 
of  country  are  usually  required. 

(c)  Method  of  construction.    It  is  generally  specified  that  the 
ties  shall  be  hewed  on  two  sides ;   that  the  two  faces  thus  made 
shall  be  parallel  planes  and  that  the  bark  shall  be  removed.     It 
is  sometimes  required  that  the  ends  shall  be  sawed  off  square ; 
that  the  timber  shall  be  cut  in  the  winter  (when  the  sap  is  down) ; 
and  that  the  ties  shall  be  seasoned  for  six  months.     These  last 
specifications  are  not  required  or  lived  up  to  as  much  as  their 
importance  deserves.    It  is  sometimes  required  that  the  ties  shall 
be  delivered  on  the  right  of  way,  neatly  piled  in  rows,  the  alter- 
nate rows  at  right  angles,  piled  if  possible  on  ground  not  lower 
than  the  rails  and  at  least  seven  feet  away  from  them,  the  lower 
row  of  ties  resting  on  two  ties  which  are  themselves  supported 
so  as  to  be  clear  of  the  ground. 

(d)  Quality  of  timber.     The  usual  specifications  for  sound 
timber  are  required,  except  that  they  are  not  so  rigid  as  for  a 
better  class  of  timber  work.     The  ties  must  be  sound,'  reason- 


§208.  ^     TIES.  231 

ably  straight-grained,  and  not  very  crooked — one  test  being  that 
a  line  joining  the  center  of  one  end  with  the  center  of  the  middle 
shall  not  pass  outside  of  the  other  end.  Splits  or  shakes,  espe- 
cially if  severe,  should  cause  rejection. 

Specifications  sometimes  require  that  the  ties  shall  be  cut 
from  single  trees,  making  what  is  known  as  "  pole  ties"  and 
definitely  condemning  those  which 
are  cut  or  split  from  larger  trunks, 


giving   two  "slab   ties"  or  four     POLET1E.     .     SLABTIE.       QUARTER TIE 
4 'quarter   ties"   for   each    cross- FIG.  109.— METHODS   OP   CUTTING 
section,   as  is  illustrated  in   Fig.  TlES- 

109.     Even  if  pole  ties  are  better,  their  exclusive  use  means  the 
rapid  destruction  of  forests  of  young  trees. 

208.  Regulations  for  laying  and  renewing  ties.  The  regula- 
tions issued  by  railroad  companies  to  their  track  foremen  will 
generally  include  the  following,  in  addition  to  directions  regard- 
ing dimensions,  spacing,  and  specifications  given  in  §§  20i— 207. 
When  hewn  ties  of  somewhat  variable  size  are  used,  as  is  fre- 
quently the  case,  the  largest  and  best  are  to  be  selected  for  use 
as  joint  ties.  If  the  upper  surface  of  a  tie  is  found  to  be  warped 
(contrary  to  the  usual  specifications)  so  that  one  or  both  rails  do 
not  get  a  full  bearing  across  the  whole  width  of  the  tie,  it  must 
be  adzed  to  a  true  surface  along  its  whole  length  and  not  merely 
notched  for  a  rail-seat.  When  respiking  is  necessary  and  spikes 
have  been  pulled  out,  the  holes  should  be  immediately  plugged 
with  "wooden  spikes,"  which  are  supplied  to  the  foremen  for 
that  express  purpose,  so  as  to  fill  up  the  holes  and  prevent  the 
decay  which  would  otherwise  take  place  when  the  hole  becomes 
filled  with  rain-water.  Ties  should  always  be  laid  at  right  angles 
to  the  rails  and  never  obliquely.  Minute  regulations  to  prevent 
premature  rejection  and  renewal  of  ties  are  frequently  made.  It 
is  generally  required  that  the  requisitions  for  renewals  shall  be 
made  by  the  actual  count  of  the  individual  ties  to  be  renewed 
instead  of  by  any  wholesale  estimates.  It  is  unwise  to  have  ties 
of  widely  variable  size,  hardness,  or  durability  adjacent  to  each 


232  RAILROAD  CONSTRUCTION.  §  209. 

other  in  the  track,  for  the  uniform  elasticity,  so  necessary  for 
smooth  riding,  will  be  unobtainable  under  those  circumstances. 

209.  Cost  of  ties.     When  railroads  can  obtain  ties  cut  by 
farmers  from  woodlands  in   the  immediate  neighborhood,  the 
price  will  frequently  be  as  low  as  20  c.  for  the  smaller  sizes, 
running  up  to  50  c.  for  the  larger  sizes  and  better  qualities,  espe- 
cially when  the  timber  is  not  very  plentiful.     Sometimes  if  a 
railroad  cannot  procure  suitable  ties  from  its  immediate  neigh- 
borhood, it  will  find  that  adjacent  railroads  control  all  adjacent 
sources  of  supply  for  their  own  use  and  that  ties  can  only  be 
procured  from  a  considerable  distance,  with  a  considerable  added 
cost  for  transportation.    First-class  oak  ties  cost  about  75  to  80  c. 
and  frequently  much  more.      Hemlock  ties  can   generally  be 
obtained  for  35  c.  or  less. 

PRESERVATIVE    PROCESSES    FOR    WOODEN    TIES. 

210.  General  principle.     "Wood  has  a  fibrous  cellular  struc- 
ture, the  cells  being  filled  with  sap  or  air.      The  woody  fiber  is 
but  little  subject  to  decay  unless  the  sap  undergoes  fermentation. 
Preservative  processes  generally  aim  at  removing  as  much  of  the 
water  and  sap  as  possible  and  filling  up  the  pores  of  the  wood 
with  an  antiseptic  compound.     The  most  common  methods  (ex- 
cept one)  all  agree  in  this  general  process  and  only  differ  in  the 
method  employed  to  get  rid   of  the  sap  and  in  the  antiseptic 
chemical  with  which  the  fibers  are  filled.      One  valuable  feature 
of  these  processes  lies  in  the  fact  that  the  softer  cheaper  woods 
(such  as  hemlock  and  pine)  are  more  readily  treated  than  are  the 
harder  woods  and  yet  will  produce  practically  as  good  a  tie  as  a 
treated   hard -wood   tie  and  a  very  much  better  tie  than  an  un- 
treated hard-wood   tie.      The  various   processes  will   be  briefly 
described,   taking  up  first  the  process  which  is   fundamentally 
different  from  the  others,  viz.,  vulcanizing. 

211.  Vulcanizing.    The  process  consists  in  heating  the  timber 
to  a  temperature  of  300°  to  500°  F.  in  a  cylinder,  the  air  being 
under  a  pressure  of  100  to  175  Ibs.   per  square  inch.      By  this 
process  the  albumen  in  the  sap  is  coagulated,  the  water  evap- 


§  212.  TIES.  233 

orated,  and  the  pores  are  partially  closed  by  the  coagulation  of 
the  albumen.  It  is  claimed  that  the  heat  sterilizes  the  wood  and 
produces  chemical  changes  in  the  wood  which  give  it  an  antisep- 
tic character.  It  has  been  very  extensively  used  on  the  elevated 
lines  of  Kew  York  City,  and  it  is  claimed  to  give  perfect  satis- 
faction. The  treatment  has  cost  that  road  25  c.  per  tie. 

212.  Creosoting.  This  process  consists  in  impregnating  the 
wood  with  wood-creosote  or  with  dead  oil  of  coal-tar.  Wood- 
creosote  is  one  of  the  products  of  the  destructive  distillation  of 
wood — usually  long-leaf  pine.  Dead  oil  of  coal-tar  is  a  prod- 
uct of  the  distillation  of  coal-tar  at  a  temperature  between  480° 
and  760°  F.  It  would  require  about  35  to  50  pounds  of  creo- 
sote to  completely  iill  the  pores  of  a  cubic  foot  of  wood.  But 
it  would  be  impossible  to  force  such  an  amount  into  the  wood, 
nor  is  it  necessary  or  desirable.  About  10  pounds  per  cubic 
foot,  or  about  35  pounds  per  tie,  is  all  that  is  necessary.  For 
piling  placed  in  salt  water  about  18  to  20  pounds  per  cubic  foot 
is  used,  and  the  timber  is  then  perfectly  protected  against  the 
ravages  of  the  teredo  navalis.  To  do  the  work,  long  cylinders, 
which  may  be  opened  at  the  ends,  are  necessary.  Usually  the 
timbers  are  run  in  and  out  011  iron  carriages  running  on  rails 
fastened  to  braces  on  the  inside  of  the  cylinder.  When  the  load 
has  been  run  in,  the  ends  of  the  cylinder  are  fastened  on.  The 
water  and  air  in  the  pores  of  the  wood  are  first  drawn  out  by 
subjecting  the  wood  alternately  to  steam-pressure  and  to  the 
action  of  a  vacuum-pump.  This  is  continued  for  several  hours. 
Then,  after  one  of  the  vacuum  periods,  the  cylinder  is  filled 
with  creosote  oil  at  a  temperature  of  about  170°  F.  The  pumps 
are  kept  at  work  until  the  pressure  is  about  80  to  100  pounds 
per  square  inch,  and  is  maintained  at  this  pressure  from  one  to 
two  hours  according  to  the  size  of  the  timber.  The  oil  is  then 
withdrawn,  the  cylinders  opened,  the  train  pulled  out  and  an- 
other load  made  up  in  40  to  60  minutes.  The  average  time  re- 
quired for  treating  a  load  is  about  18  or  20  hours,  the  absorption 
about  10  or  11  pounds  of  oil  per  cubic  foot,  and  the  cost  (1894) 
from  $12.50  to  $14.50  per  thousand  feet  B.  M. 


234  RAILROAD  CONSTRUCTION.  §213. 

213.  Burnettizing  (chloride-of-zinc  process).     This  process  is 
very  similar  to  the  creosoting  process  except  that  the  chemical  is 
chloride  of  zinc,  and  that  the  chemical  is  not  heated  before  use. 
The  preliminary  treatment  of  the  wood  to  alternate  vacuum  and 
pressure  is  not  continued  for  quite  so  long  a  period  as  in  the 
creosoting  process.     Care  must  be  taken,  in  using  this  process, 
that  the  ties  are  of  as  uniform  quality  as  possible,  for  seasoned 
ties  will  absorb  much  more  zinc  chloride  than  unseasoned  (in  the 
same  time),  and  the  product  will  lack  uniformity  unless  the  sea- 
soning is  uniform.     The  A.,  T.  &  S.  Fe  K.E.  has  works  of*  its 
own  at  which  ties  are  treated  by  this  process  at  a  cost  of  about 
25  c.  per  tie.     The  Southern  Pacific  R.R.  also  has  works  for 
burnettizing  ties  at  a  cost  of  9.5  to  12  c.   per  tie.     The  zinc- 
chloride  solution  used  in  these  works  contains  only  1.7$  of  zinc 
chloride  instead  of  over  Z%  as  used  in  the  Santa  Fe  works,  which 
perhaps  accounts  partially  for  the  great  difference  in  cost  per  tie. 
One  great  objection  to  burnettized  ties  is  the  fact  that  the  chem- 
ical is  somewhat  easily  washed  out,  when  the  wood  again  be- 
comes subject  to  decay.       Another  objection,  which  is  more 
forcible  with  respect  to  timber  subject   to  great  stresses,  as  in 
trestles,  than  to  ties,  is  the  fact  that  when  the  solution  of  zinc 
chloride  is  made  strong  (over  3$)  the  timber  is  made  very  brittle 
.•and  its  strength  is  reduced.    The  reduction  in  strength  has  been 
shown  by  tests  to  amount  to  J  to  -^  of  the  ultimate  strength, 
and  that  the  elastic  limit  has  been  reduced  by  about  \. 

214.  Kyanizing   (bichloride-of-mercury  or  corrosive-sublimate 
process) .     This  is  a  process  of  c '  steeping. ' '     It  requires  a  much 
longer  time  than  the  previously  described  processes,  but  does  not 
require  such  an  expensive  plant.     Wooden  tanks  of  sufficient 
size  for  the  timber  are  all  that  is  necessary.     The  corrosive  subli- 
mate is  first  made  into  a  concentrated  solution  of  one  part  of 
chemical  to  six  parts  of  hot  water.     When  used  in  the  tanks  this 
solution  is  weakened  to  1  part  in  100  or  150.     The  wood  will 
absorb  about  5  to  6.5  pounds  of  the  bichloride  per  100  cubic 
feet,  or  about  one  pound  for  each  4  to  6  ties.     The  timber  is 
allowed  to  soak  in  the  tanks  for  several  days,  the  general  rule 


§215.  .       TIES.  235 

being  about  one  day  for  each  inch  of  least  thickness  and  one  day 
over — which  means  seven  days  for  six-inch  ties,  or  thirteen  (to 
fifteen)  days  for  12"  timber  (least  dimension).  The  process  is 
somewhat  objectionable  on  account  of  the  chemical  being  such  a 
virulent  poison,  workmen  sometimes  being  sickened  by  the  fumes 
arising  from  the  tanks.  On  the  Baden  railway  (Germany) 
kyanized  ties  last  20  to  30  years.  On  this  railway  the  wood  is 
always  air-dried  for  two  weeks  after  impregnation  and  before 
being  used,  which  is  thought  to  have  an  important  effect  on  its 
durability.  The  solubility  of  the  chemical  and  the  liability  of 
the  chemical  washing  out  and  leaving  the  wood  unprotected  is 
an  element  of  weakness  in  the  method. 

215.  Wellhouse  (or  zinc-tannin)  process.  The  last  two 
methods  described  (as  well  as  some  others  employing  similar 
chemicals)  are  open  to  the  objection  that  since  the  wood  is  im- 
pregnated with  an  aqueous  solution,  it  is  liable  to  be  washed  out 
very  rapidly  if  the  wood  is  placed  under  water,  and  will  even 
disappear,  although  more  slowly,  under  the  action  of  moisture 
and  rain.  Several  processes  have  been  proposed  or  patented  to 
prevent  this.  Many  of  them  belong  to  one  class,  of  which  the 
Wellhouse  process  is  a  sample.  By  these  processes  the  timber 
is  successively  subjected  to  the  action  of  two  chemicals,  each 
individually  soluble  in  water,  and  hence  readily  impregnating 
the  timber,  but  the  chemicals  when  brought  in  contact  form  in- 
soluble compounds  which  cannot  be  washed  out  of  the  wood- 
cells.  By  the  Wellhouse  process,  the  wood  is  first  impregnated 
with  a  solution  of  chloride  of  zinc  and  glue,  and  is  then  subjected 
to  a  bath  of  tannin  under  pressure.  The  glue  and  tannin  com- 
bine to  form  an  insoluble  leathery  compound  in  the  cells,  which 
will  prevent  the  zinc  chloride  from  being  washed  out.  It  is 
being  used  by  the  A.,  T.  &  S.  Fe  R.K.,  their  works  being 
located  at  Las  Vegas,  [New  Mexico,  and  also  by  the  Union 
Pacific  R.R.  at  their  works  at  Laramie,  Wyo.  In  1897  Mr.  J. 
M.  Meade,  a  resident  engineer  on  the  A.,  T.  &  S.  Fe,  exhibited 
to  the  Road  masters  Association  of  America  a  piece  of  a  tie  treated 
by  this  process  which  had  been  taken  from  the  tracks  after 


236  RAILROAD   CONSTRUCTION.  §  216. 

nearly  13  years'  service.  The  tie  was  selected  at  random,  was 
taken  out  for  the  sole  purpose  of  having  a  specimen,  and  was 
still  in  sound  condition  and  capable  of  serving  many  years  longer. 
The  cost  of  the  treatment  was  then  quoted  as  13  c.  per  tie. 
It  was  claimed  that  the  treatment  trebled  the  life  of  the  tie 
besides  adding  to  its  spike-holding  power. 

216.  Cost  of  treating.     The  cost  of  treating  ties  by  the  vari- 
ous methods   has   been    estimated   as  follows  * — assuming  that 
the    plant  was  of  sufficient  capacity  to  do  the  work  economi- 
cally :    creosoting,   25  c.  per  tie ;    vulcanizing,    25   c.   per  tie ; 
burnettizing    (chloride    of   zinc),    8.25   c.    per   tie  ;    kyanizing 
(steeping  in  corrosive  sublimate),   14.6  c.   per  tie;    Wellhouse 
process  (chloride  of  zinc  and  tannin),  11.25  c.  per  tie.     These 
estimates  are  only  for  the  net  cost  at  the  works  and  do  not 
include  the  cost  of  hauling  the  ties  to  and  from  the  works,  which 
may  mean  5  to  10  c.  per  tie.      Some  of  these  processes  have 
been  installed  on  cars  which  are  transported  over  the  road  and 
operated  where  most  convenient. 

217.  Economics  of  treated  ties.     The  fact  that  treated  ties  are 
not  universally  adopted  is  due  to  the  argument  that  the  added 
life  of  the  tie  is  not  worth  the  extra  cost.     If  ties  can  be  bought 
for   25  c.,   and   cost   25   c.   for  treatment,   and   the   treatment 
only  doubles  their  life,   there   is  apparently  but  little  gained 
except  the  work  of  placing  the  extra  tie  in  the  track,  which  is 
more  or  less  offset  by  the  interest  on  25  c.  for  the  life  of  the 
untreated  tie,  and  the  larger  initial  outlay  makes  a  stronger  im- 
pression  on   the   mind   than    the   computed   ultimate   economy. 
But  when  ties    cost   75   c.    and    treatment   costs    only   25    c., 
or  perhaps  less,  then  the  economy  is  more  apparent  and  un- 
questionable.     But  this  analysis  may  be    made    more   closely. 
As  shown  in  §  202,  the  disturbance  of  the  roadbed  on  account 
of  frequent  renewals  of  untreated  ties  is  a  disadvantage  which 
would  justify  an  appreciable  expenditure  to  avoid,  although  it  is 


*Bull.  No.  9,  U.  S.  Dept.  of  Agric.,  Div.  of  Forestry.     App.  No.  1,  by 
Henry  Flad. 


§  217.  -       TIES.  237 

very  difficult  to  closely  estimate  its  true  value.  The  annual  cost 
of  a  system  of  ties  may  be  considered  as  the  sum  of  (a)  the 
interest  on  the  first  cost,  (I)  the  annual  sinking  fund  that  would 
buy  a  new  tie  at  the  end  of  its  life,  and  (c)  the  average  annual 
-cost  of  maintenance  for  the  life  of  the  tie,  which  includes  the 
-cost  of  laying  and  the  considerable  amount  of  subsequent  tamp- 
ing that  must  be  done  until  the  tie  is  fairly  settled  in  the  road- 
bed, beside  the  regular  track  work  on  the  tie,  which  is  practically 
constant.  This  last  item  is  difficult  to  compute,  but  it  is  easy  to 
see  that,  since  the  cost  of  laying  the  tie  and  the  subsequent 
tamping  to  obtain  proper  settlement  is  the  same  for  all  ties  (of 
similar  form),  the  average  annual  charge  on  the  longer-lived  tie 
would  be  much  less.  In  the  following  comparison  item  (c)  is 
disregarded,  simply  remembering  that  the  advantage  is  with  the 


longer-lived  tie. 


Untreated  tie. 


Original  cost 40  cents 

Life  (assumed  at) 7  years 


Item  (a) — interest  on  first  cost  @  4$ 1.6  cents 

"     (&)— sinking  fund  @  4# 5.1     " 

"     (c] — (considered  here  as  offsetted) . .  . 


Treated  tie. 

65  cents 
14  years 


2.6  cents 
3.6 


Average  annual  cost  (except  item  (<?))  ....   6.7  cents    6.2  cents 

On  this  basis  treated  ties  will  cost  0.5  cent  less  per  annum 
besides  the  advantage  of  item  (c}  and  the  still  more  indefinite 
advantages  resulting  from  smoother  running  of  trains,  less  wear 
and  tear  on  rolling  stock,  etc.,  due  to  less  disturbance  of  the 
roadbed. 

In  Europe,  where  wood  is  expensive,  untreated  ties  are 
seldom  used,  as  the  treatment  is  always  considered  to  be  worth 
more  than  it  costs.  The  rapid  destruction  of  the  forests  of  tim- 
ber in  this  country  is  having  the  effect  of  increasing  the  price,  so 
that  it  will  not  be  long  before  treated  ties  (or  metal  ties)  will  be 
economical  for  a  large  majority  of  the  railroads  of  the  country. 


238  RAILROAD  CONSTRUCTION.  §  218. 


METAL   TIES. 

218.  Extent  of  use.     In  1894  *  there  were  nearly  35000  miles 
of  "  metal  track  "  in  various  parts  of  the  world.     Of  this  total, 
there  were  3645  miles  of  "  longitudinals  "  (see  §  224),  found  ex- 
clusively in  Europe,  nearly  all  of  it  being  in  Germany.     There 
were  over  12000  miles  of  "bowls  and  plates  "  (see  §  223),  found 
almost  entirely  in  British  India  and  in  the  Argentine  Republic. 
The  remainder,  over  18000  miles,  was  laid  with  metal  cross-ties 
of  various  designs.     There  were  over  8000  miles  of  metal  cross- 
ties  in  Germany  alone,  about  1500  miles  in  the  rest  of  Europe, 
over  6000  miles  in  British  India,  nearly  1000  miles  in  the  rest 
of  Asia,  and  about  1500  miles  more  in  various  other  parts  of  the 
world.     Several  railroads  in  this  country  have  tried  various  de- 
signs of  these  ties,  but  their  use  has  never  passed  the  experi- 
mental stage.     These   35000  miles  represent  about  9$  of  the 
total  railroad  mileage  of  the  world—nearly  400000  miles.     They 
represent  about  17.6$  of  the  total  railroad  mileage,  exclusive  of 
the  United  States  and  Canada,  where  they  are  not  used  at  all, 
except  experimentally.     In  the  four  years  from  1890  to  1894  the 
use  of  metal  track  increased  from  less  than  25000  miles  to  nearly 
35000  miles.     This  increase  was  practically  equal  to  the  total  in- 
crease in  railroad  mileage  during  that  time,  exclusive  of  the  in- 
crease in  the  United  States  and  Canada.     This  indicates  a  large 

C5 

growth  ill  the  percentage  of  metal  track  to  total  mileage,  and 
therefore  an  increased  appreciation  of  the  advantages  to  be  de- 
rived from  their  use. 

219.  Durability.     The  durability  of  metal  track  is  still  far 
from  being  a  settled  question,  due  largely  to  the  fact  that  the 
best  form  for  such  track  is  not  yet  determined,  and  that  a  large 
part  of  the  apparent  failures  in  metal  track  have  been  evidently 
due  to  defective  design.     Those  in  favor  of  them  estimate  the 
life  as  from  30  to  50  years.     The  opponents  place  it  as  not  more 
than  20  years,  or  perhaps  as  long  as  the  best  of  wooden  ties. 

*  Bulletin  No.  9,  U.  S.  Dept.  of  Agriculture,  Div.  of  Forestry. 


§220.  ,      TIES.  239 

Unlike  the  wooden  tie,  however,  which  deteriorates  as  much 
with  time  as  with  usage,  the  life  of  a  metal  tie  is  more  largely  a 
function  of  the  traffic.  The  life  of  a  well-designed  metal  tie  has 
been  estimated  at  150000  to  200000  trains;  for  20  trains  per 
day,  or  say  6000  per  year,  this  would  mean  from  25  to  33  years. 
20  trains  per  day  on  a  single  track  is  a  much  larger  number  than 
will  be  found  on  the  majority  of  railroads.  Metal  ties  are  found 
to  be  subject  to  rust,  especially  when  in  damp  localities,  such  as 
tunnels ;  but  on  the  other  hand  it  is  in  such  confined  localities, 
where  renewals  are  troublesome,  that  it  is  especially  desirable  to 
employ  the  best  and  longest-lived  ties.  Paint,  tar,  etc.,  have 
been  tried  as  a  protection  against  rust,  but  the  efficacy  of  such 
protection  is  as  yet  uncertain,  the  conditions  preventing  any  re- 
newal of  the  protection — such  as  may  be  done  by  repainting  a 
bridge,  for  example.  Failures  in  metal  cross-ties  have  been 
largely  due  to  cracks  which  begin  at  a  corner  of  one  of  the  square 
holes  which  are  generally  punched  through  the  tie,  the  holes 
being  made  for  the  bolts  by  which  the  rails  are  fastened  to  the 
tie.  The  holes  are  generally  punched  because  it  is  cheaper. 
Reaming  the  holes  after  punching  is  thought  to  be  a  safeguard 
against  this  frequent  cause  of  failure.  Another  method  is  to 
round  the  corners  of  the  square  punch  with  a  radius  of  about 
•J".  If  a  crack  has  already  started,  the  spread  of  the  crack  may 
be  prevented  by  drilling  a  small  hole  at  the  end  of  it. 

220.  Form  and  dimensions  of  metal  cross-ties.  Since  stability 
in  the  ballast  is  an  essential  quality  for  a  tie,  this  must  be  accom- 
plished either  by  turning  down  the  end  of  the  tie  or  by  having 
some  form  of  lug  extending  downward  from  one  or  more  points 
of  the  tie.  The  ties  are  sometimes  depressed  in  the  center  (see 
Plate  XVII,  K  Y.  C.  &  H.  E.  R.E.  tie)  to  allow  for  a  thick 
covering  of  ballast  on  top  in  order  to  increase  its  stability  in  the 
ballast,  This  form  requires  that  the  ties  should  be  sufficiently 
well  tamped  to  prevent  a  tendency  to  bend  out  straight,  thus 
widening  the  gauge.  Many  designs  of  ties  are  objection- 
able because  they  cannot  be  placed  in  the  track  without 
disturbing  adjacent  ties.  The  failure  of  many  metal  cross- 


240  RAILROAD   CONSTRUCTION.  §  221. 

ties,  otherwise  of  good  design,  may  be  ascribed  to  too  light 
weight.  Those  weighing  much  less  than  100  pounds  have 
proved  too  light.  From  100  to  130  pounds  weight  is  being  used 
satisfactorily  on  German  railroads.  The  general  outside  dimen- 
sions are  about  the  same  as  for  wooden  ties,  except  as  to  thick- 
ness. The  metal  is  generally  from  J"  to  f "  thick.  They  are, 
of  course,  only  made  of  wrought  iron  or  steel,  cast  iron  being 
used  only  for  "  bowls  "  or  "  plates  "  (see  §  213).  The  details 
of  construction  of  some  of  the  most  commonly  used  ties  may  be 
seen  by  a  study  of  Plate  XVII. 

221.  Fastenings.     The  devices  for  fastening  the  rails  to  the 
ties  should  be  such  that  the  gauge  may  be  widened  if  desired  on 
curves,  also  that  the  gauge  can  be  made  true  regardless  of  slight 
inaccuracies  in  the  manufacture  of  the  ties,  and  also  that  shims 
may  be  placed  under  the  rail  if  necessary  during  cold  weather 
when   the  tie  is  frozen  into  the  ballast  and  cannot  be  easily 
disturbed.     Some  methods  of  fastening  require  that  the  base  of 
the  rail  be  placed  against  a  lug  which  is  riveted  to  the  tie  or 
which  forms  a  part  of  it.      This  has  the  advantage  of  reducing 
the  number  of  pieces,  but  is  apt  to  have   one  or  more  of  the 
disadvantages  named  above.     Metal  keys  or  wooden  wedges  are 
sometimes  used,  but  the  majority  of  designs  employ  some  form 
of  bolted  clamp.     The  form  adopted  for  the  experimental  ties 
used  by  the  N.  Y.  C.  &  H.  K.  R.R.  (see  Plate  XVII)  is  especially 
ingenious  in  the  method  used  to  vary  the  gauge   or  allow  for 
inaccuracies  of  manufacture.     Plate  XVII  shows  some  of  the 
methods  of  fastening  adopted  on  the  principal  types  of  ties. 

222.  Cost.     The  cost  of  metal  cross-ties  in  Germany  averages 
about  1.6  c.  per  pound  or  about  $1.60  for  a  100-lb.  tie.     The 
ties    manufactured   for  the  N.  Y.  C.    &  H.  R.  R.R.    in  1892 
weighed  about  100  Ibs.  and  cost  $2.50  per  tie,  but  if  they  had 
been  made  in   larger  quantities   and  with  the  present  price  of 
steel  the    cost   would  possibly   have    been   much   lower.     The 
item  of  freight  from  the  place  of  manufacture  to  the  place  where 
used  is  no  inconsiderable  item  of  cost  with  some  roads.      Metal 
cross-ties  have  been  used  by  some  street  railroads  in  this  country. 


PLATE    XVII. 


METAL  TIES. 


(Tofatf  pnge  340 


OF  THK 

UNIVERSITY 


§  224.  ,    TIES.  241 

Those  used  on  the  Terre  Haute  Street  Kailway  weigh  60  pounds 
and  cost  about  66  c.  for  the  tie,  or  74  c.  per  tie  with  the 
fastenings. 

223.  Bowls  or  plates.  As  mentioned  before,  over  12000 
miles  of  railway,  chiefly  in  British  India  and  in  the  Argentine 
Republic,  are  laid  with  this  form  of  track.  It  consists  essentially 
of  large  cast-iron  inverted  ' '  bowls ' '  laid  at  intervals  under  each 
rail  and  opposite  each  other,  the  opposite  bowls  being  tied 
together  with  tie-rods.  A  suitable  chair  is  riveted  or  bolted  on 
to  the  top  of  each  bowl  so  as  to  properly  hold  the  rail.  Being 
made  of  cast  iron,  they  are  not  so  subject  to  corrosion  as  steel 
or  wrought  iron.  They  have  the  advantage  that  when  old  and 
worn  out  their  scrap  value  is  from  60  to  80$  of  their  initial 
cost,  while  the  scrap  value  of  a  steel  or  wrought-iron  tie  is 
practically  nothing.  Failure  generally  occurs  from  breakage, 
the  failures  from  this  cause  in  India  being  about  0.4  per  cent 
per  annum.  They  weigh  about  250  Ibs.  apiece  and  are  there- 
fore quite  expensive  in  first  cost  and  transportation  charges. 
There  are  miles  of  them  in  India  which  have  already  lasted 
25  years  and  are  still  in  a  serviceable  condition.  Some  illustra- 
tions of  this  form  of  tie  are  shown  in  Plate  XVII. 


224.  Longitudinals.*  This  form,  the  use  of  which  is  con- 
£ned  almost  exclusively  to  Germany,  is  being  gradually  replaced 
on  many  lines  by  metal  cross-ties.  The  system  generally  con- 
sists of  a  compound  rail  of  several  parts,  the  upper  bearing  rail 
Leing  very  light  and  supported  throughout  its  length  by  other 
rails,  which  are  suitably  tied  together  with  tie-rods  so  as  to 
maintain  the  proper  gauge,  and  which  have  a  sufficiently  broad 

*  Although  the  discussion  of  longitudinals  might  be  considered  to  belong 
more  properly  to  the  subject  of  RAILS,  yet  the  essential  idea  of  all  designs 
must  necessarily  be  the  support  of  a  rail-head  on  which  the  rolling  stock  may 
run,  and  therefore  this  form,  unused  in  this  country,  will  be  briefly  described 
here. 


242  RAILROAD  CONSTRUCTION.  §  224. 

base  to  be  properly  supported  in  the  ballast.  One  great  objection 
to  this  method  of  construction  is  the  difficulty  of  obtaining 
proper  drainage  especially  on  grades,  the  drainage  having  a 
tendency  to  follow  along  the  lines  of  the  rails. 
The  construction  is  much  more  complicated  on 
sharp  curves  and  at  frogs  and  switches.  An- 
other fundamentally  different  form  of  longi- 
FIQ.  110.  tudinal  is  the  Haarman  compound  "self -bear- 
ing" rail,  having  a  base  12"  wide  and  a  height  of  8",  the 
alternate  sections  breaking  joints  so  as  to  form  a  practically 
continuous  rail. 

Some  of  the  other  forms  of  longitudinals  are  illustrated  in 
Plate  XVII. 

For  a  very  complete  discussion  of  the  subject  of  metal  ties, 
see  the  ' '  Report  on  the  Substitution  of  Metal  for  Wood  in 
Kailroad  Ties"  by  E.  E.  Russell  Tratman,  it  being  Bulletin 
No.  4,  Forestry  Division  of  the  U.  S.  Dept.  of  Agriculture. 


CHAPTER  IX. 


RAILS. 

225.  Early  forms.  The  first  rails  ever  laid  were  wooden 
stringers  which  were  used  on  very  short  train-roads  around  coal- 
mines. As  the  necessity  for  a  more  durable  rail  increased, 
owing  chiefly  to  the  invention  of  the  locomotive  as  a  motive 
power,  there  were  invented  successively  the  cast-iron  "  fish- 
belly  "  rail  and  various  forms  of  wrought-iron  strap  rails  which 
finally  developed  into  the  T  rail  used  in  this  country  and  the 
double-headed  rail,  supported  by  chairs,  used  so  extensively  in 
England.  The  cast-iron  rails  were  cast  in  lengths  of  about  3 
feet  and  were  supported  in  iron  chairs  which  were  sometimes 
set  upon  stone  piers.  A  great  deal  of  the  first  railroad  track 
of  this  country  was  laid  with  longitudinal  stringers  of  wood 
placed  upon  cross-ties,  the  inner  edge  of  the  stringers  being 


BALT.  4  OHIO  R.  R. 

QUINCYR.  R.  1843.  "BULL-HEAD." 

1826. 


'FISH-BELLY"  — CAST    IRON. 


VIGNOLES.      1836. 


JL 


,- , .J-*, ^— , _•-*_-_  - 


CAMDEN  &  AMBOY.         STEPHENSON.          ' '  PEAR.  •• 

1832.  1838.  REYNOJ-DS-1767. 

FIG.  111.— EARLY  FORMS  OF  RAILS. 

protected  by  wrought-iron   straps.     The   "bridge"  rails  were 
first  rolled  in  this  country  in  1844.     The  "pear"  section  was 

243 


244  RAILROAD  CONSTRUCTION.  §  226. 

an  approacli  to  the  present  form,  but  was  very  defective  on 
account  of  the  difficulty  of  designing  a  good  form  of  joint.  The 
"Stevens"  section  was  designed  in  1830  by  Col.  Robert  L. 
Stevens,  Chief  Engineer  of  the  Cam  den  and  Amboy  Railroad  ; 
although  quite  defective  in  its  proportions,  according  to  the 
present  knowledge  of  the  requirements,  it  is  essentially  the  pres- 
ent form.  In  1836,  Charles  Yignoles  invented  essentially  the 
same  form  in  England  ;  this  form  is  therefore  known  throughout 
England  and  Europe  as  the  Vignoles  rail. 

226,  Present  standard  forms,  The  larger  part  of  modern 
railroad  track  is  laid  with  rails  which  are  either  "  T  "  rails  or 
the  double-headed  or  "  bull  -headed  "  rails  which  are  carried  in 
chairs.  The  double-headed  rail  was  designed  with  a  symmetri- 
cal form  with  the  idea  that  after  one  head  had  been  worn  out 
by  traffic  the  rail  could  be  reversed,  and  that  its  life  would  be 
practically  doubled.  Experience  has  shown  that  the  wear  of  the 
rail  in  the  chairs  is  very  great  ;  so  much  so  that  when  one  head 
has  been  worn  out  by  traffic  the  whole  rail  is  generally  useless. 
If  the  rail  is  turned  over,  the  worn  places,  caused  by  the  chairs, 
make  a  rough  track  and  the  rail  appears  to  be  more  brittle  and 
subject  to  fracture,  possibly  due  to  the  crystallization  that  may 
have  occurred  during  the  previous  usage  and  to  the  reversal  of 
stresses  in  the  fibers.  Whatever  the  explanation,  experience  has 
demonstrated  the  fact.  The  "  bull-headed  "  rail  has  the  lower 
head  only  large  enough  to  properly  hold 
the  wooden  keys  with  which  the  rail  is 
secured  to  the  chairs  (see  Fig.  112)  and 
furnish  the  necessary  strength.  The  use 

FIG.  11  2.—  BULL  HEADED  of  these  rails  requires  the  use  of  two  cast- 
RAIL  AND  CHAIR.  It  ig  c]aimed  that 


such  track  is  better  for  heavy  and  fast  traffic,  but  it  is  more 
expensive  to  build  and  maintain.  It  is  the  standard  form  of 
track  in  England  and  some  parts  of  Europe. 

Until  a  few  years  ago  there  was  a  very  great  multiplicity 
in  the  designs  of  "T"  rails  as  used  in  this  country,  nearly 
every  prominent  railroad  having  its  own  special  design,  which 


§226. 


OF  THE 

RSITT 


RAILS. 


perhaps  differed  from  that  of  some  other  road  by  only  a  very 
minute  and  insignificant  detail,  but  which  nevertheless  would 
require  a  complete  new  set  of  rolls  for  rolling.  This  certainly 
must  have  had  a  very  appreciable  effect  on  the  cost  of  rails.  In 
1893,  the  American  Society  of  Civil  Engineers,  after  a  very 
exhaustive  investigation  of  the  subject,  extending  over  several 
years,  having  obtained  the  opinions  of  the  best  experts  of  the 
country,  adopted  a  series  of  sections  which  have  been  very  ex- 
tensively adopted  by  the  railroads  of  this  country.  Instead  of 
having  the  rail  sections  for  various  weights  to  be  geometrically 
similar  figures,  certain  dimensions  are  made  constant,  regardless 
of  the  weight.  It  was  decided  that  the  metal  should  be  dis- 
tributed through  the  section  in  the  proportions  of — head  42$, 
web  21$,  and  flange  37$.  The  top  of  the  head  should  have  a 
radius  of  12" ;  the  top  corner  radius  of  head  should  be  £%" ;  the 


FIG.  113.— AM.  Soc.  C.  E.  STANDARD  RAIL  SECTION. 

lower  corner  radius  of  head  should  be  TV' ;  the  corners  of  the 
flanges,  -fa"  radius;  side  radius  of  web,  12";  top  and  bottom 
radii  of  web  corners,  J" ;  and  angles  with  the  horizontal  of  the 
under  side  of  the  head  and  the  top  of  the  flange,  13°.  The 
sides  of  the  head  are  vertical. 

The  height  of  the  rail  (D)  and  the  width  of  the  base  (C)  are 
always  made  equal  to  each  other. 


246 


RAILROAD  CONSTRUCTION. 


227. 


Weight  per  Yard. 

40 

45 

50 

55 

60 

65 

70 

75 

80 

85 

90 

95 

100 

A 
B 
C&D 
E 
F 
G 

11" 

Si 

3* 

i 
iff 
V* 

2" 

u 

3H 
§4 

m 
i.4k 

24" 

T7B 

3S 

H 

** 

H 

2J" 

4! 

SI 

24J 

HI 

2|" 
tt 

4i  !• 

IS 
241 
i& 

241" 

\ 
4& 

il 

2| 

1/f 

2Ty 
II 

41 

M 

24S 
H4 

241" 

H 

*» 

§1 

2|f 

HI 

8|" 

11 

5 

1 

2| 

H 

2*" 
1% 
5& 

II 
2| 

Hi 

21" 
& 
5| 
1! 
2f| 

Hi 

214" 
& 
5A 

H 

2|| 
Hi 

2f' 

T9S 

H 

S^ 

3B\ 

HI 

The  chief  features  of  disagreement  among  railroad  men 
relate  to  the  radius  of  the  upper  corner  of  the  head  and  the 
slope  of  the  side  of  the  head.  The  radius  (TV')  adopted  for 
the  upper  corner  (constant  for  all  weights)  is  a  little  more 
than  is  advocated  by  those  in  favor  of  u  sharp  corners " 
who  often  use  a  radius  of  J".  On  the  other  hand  it  is 
much  less  than  is  advocated  by  those  who  consider  that  it 
should  be  nearly  equal  to  (or  even  greater  than)  the  larger 
radius  universally  adopted  for  the  corner  of 
the  wheel-flange.  The  discussion  turns  on 
the  relative  rapidity  of  rail  wear  and  the  wear 
of  the  wheel-flanges  as  affected  by  the  relation 
of  the  form  of  the  wheel-tread  to  that  of  the 
rail.  It  is  argued  that  sharp  rail  corners  wear 
the  wheel-flanges  so  as  to  produce  sharp 
flanges,  which  are  liable  to  cause  derailment 
at  switches  and  also  to  require  that  the  tires 
of  engine-drivers  must  be  more  frequently 
turned  down  to  their  true  form.  On  the 
other  hand  it  is  generally  believed  that  rail  wear  is  much  less 
rapid  while  the  area  of  contact  between  the  rail  and  wheel-flange 
is  small,  and  that  when  the  rail  has  worn  down,  as  it  invariably 
does,  to  nearly  the  same  form  as  the  wheel-flange,  the  rail  wears 
away  very  quickly. 

227.  Weight  for  various  kinds  of  traffic.  The  heaviest  rails 
in  regular  use  weigh  100  Ibs.  per  yard,  and  even  these  are  only 
used  on  some  of  the  heaviest  traffic  sections  of  such  roads  as  the 
K  Y.  Central,  the  Pennsylvania,  the  K  Y.,  N.  H.  &  H.,  and 


FIG.  114.  —  RELA- 
TION OF  RAIL  TO 
WHEEL-TREAD. 


§  228.  RAILS.  247 

a  few  others.  Probably  the  larger  part  of  the  mileage  of  the 
-country  is  laid  with  60-  to  75-lb.  rails — considering  the  fact  that 
4 '  the  larger  part  of  the  mileage ' '  consists  of  comparatively 
light-traffic  roads  and  may  exclude  all  the  heavy  trunk  lines. 
Yery  light-traffic  roads  are  sometimes  laid  with  56-lb.  rails. 
Koads  with  fairly  heavy  traffic  generally  use  75-  to  85-lb.  rails, 
especially  when  grades  are  heavy  and  there  is  much  and  sharp 
curvature.  The  tendency  on  all  roads  is  toward  an  increase  in 
the  weight,  rendered  necessary  on  account  of  the  increase  in  the 
weight  and  capacity  of  rolling  stock,  and  due  also  to  the  fact  that 
the  price  of  rails  has  been  so  reduced  that  it  is  both  better  and 
cheaper  to  obtain  a  more  solid  and  durable  track  by  increasing 
the  weight  of  the  rail  rather  than  by  attempting  to  support  a 
weak  rail  by  an  excessive  number  of  ties  or  by  excessive  track 
labor  in  tamping.  It  should  be  remembered  that  in  buying  rails 
the  mere  weight  is,  in  one  sense,  of  no  importance.  The  im- 
portant thing  to  consider  is  the  STRENGTH  and  the  STIFFNESS.  If 
we  assume  that  all  weights  of  rails  have  similar  cross-sections 
(which  is  nearly  although  not  exactly  true),  then,  since  for  beams 
of  similar  cross- sections  the  strength  varies  as  the  cube  of  the 
homologous  dimensions  and  the  stiffness  as  the  fourth  %)ower, 
while  the  area  (and  therefore  the  weight  per  unit  of  length) 
only  varies  as  the  square,  it  follows  that  the  stiffness  varies  as 
the  square  of  the  weight,  and  the  strength  as  the  f  power  of  the 
weight.  Since  for  ordinary  variations  of  weight  the  price  per 
ton  is  the  same,  adding  (say)  10$  to  the  weight  (and  cost)  adds 
21$  to  the  stiffness  and  over  15$  to  the  strength.  As  another 
illustration,  using  an  80-lb.  rail  instead  of  a  75-lb.  rail  adds  only 
<>f$  to  the  cost,  but  adds  about  14$  to  the  stiffness  and  nearly 
11$  to  the  strength.  This  shows  why  heavier  rails  are  more 
economical  and  are  being  adopted  even  when  they  are  not  abso- 
lutely needed  on  account  of  heavier  rolling  stock.  The  stiffness, 
strength,  and  consequent  durability  are  increased  in  a  much 
greater  ratio  than  the  cost. 

228,  Effect  of  stiffness  on  traction.     A  very  important  but 
generally  unconsidered  feature  of  a  stiff  rail  is  its  effect  on  trac- 


248  RAILROAD  CONSTRUCTION.  §  229. 

tive  force.  An  extreme  illustration  of  this  principle  is  seen 
when  a  vehicle  is  drawn  over  a  soft  sandy  road.  The  constant 
compression  of  the  sand  in  front  of  the  wheel  has  virtually  the 
same  effect  on  traction  as  drawing  the  wheel  up  a  grade  whose 
steepness  depends  on  the  radius  of  the  wheel  and  the  depth  of 
the  rut.  On  the  other  hand,  if  a  wheel,  made  of  perfectly 
elastic  material,  is  rolled  over  a  surface  which,  while  supported 
with  absolute  rigidity,  is  also  perfectly  elastic,  there  would  be  a 
forward  component,  caused  by  the  expanding  of  the  compressed 
metal  just  behind  the  center  of  contact,  which  would  just  bal- 
ance the  backward  component.  If  the  rail  wa<:  supported 
throughout  its  length  by  an  absolutely  rigid  support,  the  high 
elasticity  of  the  wheel-tires  and  rails  would  reduce  this  form  of 
resistance  to  an  insignificant  quantity,  but  the  ballast  and  even 
the  ties  are  comparatively  inelastic.  When  a  weak  rail  yields, 
the  ballast  is  more  or  less  compressed  or  displaced,  and  even 
though  the  elasticity  of  the  rail  brings  it  back  to  nearly  its 
former  place,  the  work  done  in  compressing  an  inelastic  material 
is  wholly  lost.  The  effect  of  this  on  the  fuel  account  is  certainly 
very  considerable  and  yet  is  frequently  entirely  overlooked.  It 
is  practically  impossible  to  compute  the  saving  in  tractive  power, 
and  therefore  in  cost  of  fuel,  resulting  from  a  given  increase  in 
the  weight  and  stiffness  of  the  rail,  since  the  yielding  of  the  rail 
is  so  dependent  on  the  spacing  of  the  ties,  the  tamping,  etc.  But 
it  is  not  difficult  to  perceive  in  a  general  way  that  such  an  econ- 
omy is  possible  and  that  it  should  not  be  neglected  in  considering 
the  value  of  stiffness  in  rails. 

229,  Length  of  rails.  The  standard  length  of  rails  with  most 
railroads  is  30  feet.  In  recent  years  many  roads  have  been  try- 
ing 45-foot  and  even  60-foot  rails.  The  argument  in  favor  of 
longer  rails  is  chiefly  that  of  the  reduction  in  track-joints,  which 
are  costly  to  construct  and  to  maintain  and  are  a  fruitful  source 
of  accidents.  Mr.  Morrison  of  the  Lehigh  Valley  K.R.*  de- 
clares that,  as  a  result  of  extensive  experience  with  45-foot  rails 

Report,  Headmasters  Association,  1895. 


§  230.  RAILS.  249 

on  that  road,  he  finds  that  they  are  much  less  expensive  to 
handle,  and  that,  being  so  long,  they  can  be  laid  around  sharp 
curves  without  being  curved  in  a  machine,  as  is  necessary  with 
the  shorter  rails.  The  great  objection  to  longer  rails  lies  in  the 
difficulty  in  allowing  for  the  expansion,  which  will  require,  in 
the  coldest  weather,  an  opening  at  the  joint  of  nearly  {• "  for  a 
60-foot  rail.  The  Pennsylvania  R.R.  and  the  Korfolk  and 
Western  R.R.  each  have  a  considerable  mileage  laid  with  60-foot 
rails. 

230.  Expansion  of  rails.  Steel  expands  at  the  rate  of  .0000065 
of  its  length  per  degree  Fahrenheit.  The  extreme  range  of  tem- 
perature to  which  any  rail  will  be  subjected  will  be  about  1 60°, 
or  say  from  —  20°  F.  to  +  140°  F.  With  the  above  coefficient 
and  a  rail  length  of  60  feet  the  expansion  would  be  0.0624  foot, 
or  about  f  inch.  But  it  is  doubtful  whether  there  would  ever 
be  such  a  range  of  motion  even  if  there  were  such  a  range  of 
temperature.  Mr.  A.  Torrey,  chief  engineer  of  the  Mich. 
Cent.  R.R.,  experimented  with  a  section  over  500  feet  long, 
which,  although  not  a  single  rail,  was  made  "  continuous  "  by 
rigid  splicing,  and  he  found  that  there  was  no  appreciable  addi- 
tional contraction  of  the  rail  at  any  temperature  below  +  20  F. 
The  reason  is  not  clear,  but  ike  fact  is  undeniable. 

The  heavy  girder  rails,  used  by  the  street  railroads  of  the 
country,  are  bonded  together  with  perfectly  tight  rigid  joints 
which  do  not  permit  expansion.  If  the  rails  are  laid  at  a  tem- 
perature of  60°  F.  and  the  temperature  sinks  to  0°,  the  rails 
have  a  tendency  to  contract  .00039  of  their  length.  If  this 
tendency  is  resisted  by  the  friction  of  the  pavement  in  which  the 
rails  are  buried,  it  only  results  in  a  tension  amounting  to  .00039 
of  the  modulus  of  elasticity,  or  say  10920  pounds  per  square 
inch,  assuming  28  OOP  OOP  as  the  modulus  of  elasticity.  This 
stress  is  not  dangerous  and  may  be  permitted.  If  the  tempera- 
ture rises  to  120°  F.,  a  tendency  to  expansion  and  buckling  will 
take  place,  which  will  be  resisted  as  before  by  the  pavement, 
and  a  compression  of  10920  pounds  per  square  inch  will  be  in- 
duced, which  will  likewise  be  harmless.  The  range  of  tempera- 


250 


RAILROAD   CONSTRUCTION. 


231 


ture  of  rails  which  are  buried  in  pavement  is  much  less  than 
when  they  are  entirely  above  the  ground  and  will  probably 
never  reach  the  above  extremes.  Rails  supported  on  ties  which 
are  only  held  in  place  by  ballast  must  be  allowed  to  expand  and  con- 
tract almost  freely,  as  the  ballast  cannot  be  depended  on  to  resist 
the  distortion  induced  by  any  considerable  range  of  temperature, 
especially  on  curves. 

231.  Kules  for  allowing  for  temperature.  Track  regulations 
generally  require  that  the  track  foremen  shall  use  iron  (not 
wooden)  shims  for  placing  between  the  ends  of  the  rails  while 
splicing  them.  The  thickness  of  these  shims  should  vary  with 
the  temperature.  Some  roads  use  such  approximate  rules  as  the 
following  :  "  The  proper  thickness  for  coldest  weather  is  -f$  of  an 
inch ;  during  spring  and  fall  use  i  of  an  inch,  and  in  the  very 
hottest  weather  y1^  of  an  inch  should  be  allowed."  This  is  on 
the  basis  of  a  30-foot  rail.  When  a  more  accurate  adjustment 
than  this  is  desired,  it  may  be  done  by  assuming  some  very  high 
temperature  (120°  to  150°  F.)  as  a  maximum,  when  the  joints 
should  be  tight;  then  compute  in  tabular  form  the  spacing  for 
each  temperature,  varying  by  20°,  allowing  0".0468  (almost 
exactly  -fa")  for  each  20°  change.  Such  a  tabular  form  would 
be  about  as  follows  (rail  length  30  feet) : 


Temperature  

150° 

130° 

110° 

90° 

70° 

50° 

30° 

10° 

-  10° 

-  30° 

Rail  opening.  .  . 

0 

A" 

A" 

A" 

A" 

U" 

A" 

tr 

f" 

27" 

6¥ 

One  practical  difficulty  in  the  way  of  great  refinement  in  this 
work  is  the  determination  of  the  real  temperature  of  the  rail 
when  it  is  laid.  A  rail  lying  in  the  hot  sun  has  a  very  much 
higher  temperature  than  the  air.  The  temperature  of  the  rail 
cannot  be  obtained  even  by  exposing  a  thermometer  directly  to 
the  sun,  although  such  a  result  might  be  the  best  that  is  easily 
obtainable.  On  a  cloudy  or  rainy  day  the  rail  has  practically 
the  same  temperature  as  the  air;  therefore  on  such  days  there 
need  be  no  such  trouble. 


§232.  ,     BAILS.  251 

232.  Chemical  composition.  About  98  to  99.5$  of  the  com- 
position of  steel  rails  is  iron,  but  the  value  of  the  rail,  as  a  rail, 
is  almost  wholly  dependent  upon  the  large  number  of  other 
chemical  elements  which  are,  or  may  be,  present  in  very  small 
amounts.  The  manager  of  a  steel- rail  mill  once  declared  that 
their  aim  was  to  produce  rails  having  in  them — 

Carbon 0.32  to  0.40$ 

Silicon 0.04:  to  0.06$ 

Phosphorus 0.09  to  0.105$ 

Manganese 1.00  to  1.50$ 

The  analysis  of  32  specimens  of  rails  on  the  Chic.,  Mil.  & 
St.  Paul  R.  E-.  showed  variations  as  follows : 

Carbon 0.211  to  0.52$ 

Silicon 0.013  to  0.256$ 

Phosphorus 0.055  to  0.181$ 

Manganese , 0.35    to  1.63$ 

These  quantities  have  the  same  general  relative  proportions 
as  the  rail-mill  standard  given  above,  the  differences  lying 
mainly  in  the  broadening  of  the  limits.  Increasing  the  percent- 
age of  carbon  by  even  a  few  hundredths  of  one  per  cent  makes 
the  rail  harder,  but  likewise  more  brittle.  If  a  track  is  well 
ballasted  and  not  subject  to  heaving  by  frost,  a  harder  and  more 
brittle  rail  may  be  used  without  excessive  danger  of  breakage, 
and  such  a  rail  will  wear  much  longer  than  a  softer  tougher 
rail,  although  the  softer  tougher  rail  may  be  the  better  rail  for 
a  road  having  a  less  perfect  roadbed. 

A  small  but  objectionable  percentage  of  sulphur  is  some-^ 
times  found  in  rails,  and  very  delicate  analysis  will  often  show 
the  presence,  in  very  minute  quantities,  of  several  other 
chemical  elements.  The  use  of  a  very  small  quantity  of  nickel 
or  aluminum  has  often  been  suggested  as  a  means  of  producing 
a  more  durable  rail.  The  added  cost  and  the  uncertainty  of 


252  RAILROAD  CONSTRUCTION.  §  233. 

the  amount  of  advantage  to  be  gained  has  hitherto  prevented 
the  practical  use  or  manufacture  of  such  rails. 

233.  Testing.     Chemical    and   mechanical    testing  are  both 
necessary  for  a  thorough  determination  of  the  value  of  a  rail. 
The  chemical  testing  has  for  its  main  object  the  determination 
of  those  minute  quantities  of  chemical  elements  which  have  such 
a  marked  influence  on  the  rail  for  good  or  bad.    The  mechanical 
testing  consists  of  the  usual  tests    for   elastic   limit,    ultimate 
strength,  and  elongation  at  rupture,  determined  from  pieces  cut 
out  of  the  rail,  besides  a  "  drop  test."     The  drop  test  consists 
in  dropping  a  weight  of  2000  Ibs.  from  a  height  of  16  to  20 
feet  on  to  the  center  of  a  rail  which  is  supported  on  abutments 
placed  three  or  four  feet  apart.     The  number  of  blows  required 
to  produce  rupture  or  to  produce  a  permanent  set  of  specified 
magnitude  gives  a  measure  of  the  strength  and  toughness  of 
the  rail. 

234.  Rail  wear  on  tangents.     When  the  wheel  loads  on  a  rail 
are  abnormally  heavy,   and  particularly  when  the  rail  has.  but 
little  carbon  and  is  unusually  soft,  the  concentrated  pressure  on 

the  rail  is  .frequently  greater  than  the  elastic 
limit,  and  the  metal  "  flows  "  so  that  the  head, 
although  greatly  abraded,  will  spread  somewhat 
outside  of  its  original  lines,  as  shown  in  Fig. 
115.  The  rail  wear  that  occurs  on  tangents  is 
FIG.  115.  almost  exclusively  on  top.  Statistics  show  that 

the  rate  of  rail  wear  on  tangents  decreases  as  the  rails  are  more 
worn.  Tests  of  a  large  number  of  rails  on  tangents  have  shown 
a  rail  wear  averaging  nearly  one  pound  per  yard  per  10  000  000 
tons  of  traffic.  There  is  about  33  pounds  of  metal  in  one  yard 
of  the  head  of  an  80-lb.  rail.  As  an  extreme  value  this  may  be 
•worn  down  one-half,  thus  giving  a  tonnage  of  165  000  000  tons 
for  the  life  of  the  rail.  Other  estimates  bring  the  tonnage 
down  to  125  000  000  tons.  Since  the  locomotive  is  considered 
to  be  responsible  for  one  half  (and  possibly  more)  of  the  damage 
done  to  the  rail,  it  is  found  that  the  rate  of  wear  on  roads  with 
shorter  trains  is  more  rapid  in  proportion  to  the  tonnage,  and  it 


§  235.  RAILS.  253 

is  therefore  thought  that  the  life  of  a  rail  should  be  expressed  in 
terms  of  the  number  of  trains.  This  has  been  estimated  at 
300  000  to  500  000  trains. 

235.  Rail  wear  on  curves.  On  curves  the  maximum  rail  wear 
occurs  on  the  inner  side  of  the  head  of  the  outer  rail,  giving  a 
worn  form  somewhat  as  shown  in  Fig.  116.  The  dotted  line 
shows  the  nature  and  progress  of  the  rail  wear 
on  the  inner  rail  of  a  curve.  Since  the  pressure 
on  the  outer  rail  is  somewhat  lateral  rather  than 
vertical,  the  ' 4  flow  ' '  does  not  take  place  to  the 
same  extent,  if  at  all,  on  the  outside,  and  what- 
ever flow  would  take  place  on  the  inside  is  FIG.  116. 
immediately  worn  oft'  by  the  wheel-flange..  Unlike  the  wear  on 
tangents,  the  wear  on  curves  is  at  a  greater  rate  as  the  rail 
becomes  more  worn. 

The  inside  rail  on  curves  wears  chiefly  on  top,  the  same  as 
on  a  tangent,  except  that  the  wear  is  much  greater  owing  to  the 
longitudinal  slipping  of  the  wheels  on  the  rail,  and  the  lateral 
slipping  that  must  occur  when  a  rigid  four-wheeled  truck  is 
guided  around  a  curve.  The  outside  rail  is  subjected  to  a 
greater  or  less  proportion  of  the  longitudinal  slipping,  likewise 
to  the  lateral  slipping,  and,  worst  of  all,  to  the  grinding  action 
of  the  flange  of  the  wheel,  which  grinds  off  the  side  of  the 
head. 

The  results  of  some  very  elaborate  tests,  made  by  Mr. 
A.  M.  Wellington,  on  the  Atlantic  and  Great  Western  K.E.,  on 
the  wear  of  rails,  seem  to  show  that  the  rail  wear  on  curves 
may  be  expressed  by  the  formula :  c  i  Total  wear  of  rails  on  a  d 
degree  curve  in  pounds  per  yard  per  10  000  000  tons  duty 
=  1  +  0.03<T."  "It  is  not  pretended  that  this  formula  is 
strictly  correct  even  in  theory,  but  several  theoretical  consider- 
ations indicate  that  it  may  be  nearly  so."  According  to  this 
formula  the  average  rail  wear  on  a  6°  curve  will  be  about  twice 
the  rail  wear  on  a  tangent.  While  this  is  approximately  true, 
the  various  causes  modifying  the  rate  of  rail  wear  (length  of 
trains,  age  and  quality  of  rails,  etc.)  will  result  in  numerous  and 


254  RAILROAD   CONSTRUCTION.  §  236. 

large  variations  from  the  above  formula,  which  should  only  be 
taken  as- indicating  an  approximate  law. 

236.  Cost  of  rails.  In  1873  the  cost  of  steel  rails  was  about 
$120  per  ton,  and  the  cost  of  iron  rails  about  $70  per  ton. 
Although  the  steel  rails  were  at  once  recognized  as  superior  to 
iron  rails  on  account  of  more  uniform  wear,  they  were  an 
expensive  luxury.  The  manufacture  of  steel  rails  by  the  Bes- 
semer process  created  a  revolution  in  prices,  and  they  have 
steadily  dropped  in  price  until,  during  the  last  few  years,  steel 
rails  have  been  manufactured  and  sold  for  $22  per  ton.  At 
such  prices  there  is  no  longer  any  demand  for  iron  rails,  since 
the  cost  of  manufacturing  them  is  substantially  the  same  as  that 
of  steel  rails,  while  their  durability  is  unquestionably  inferior  to 
that  of  steel  rails. 


CHAPTER  X. 
KAIL-FASTENINGS. 

RAIL-JOINTS. 

237.  Theoretical  requirements  for  a  perfect  joint.     A  perfect 
rail- joint  is  one  that  has  the  same  strength  and  stiffness — no  more 
and  no  less — as  the  rails  which  it  joins,  and  which  will  not 
interfere  with  the  regular   and  uniform    spacing  of   ties.     It 
should  also  be  reasonably  cheap  both  in  first  cost  and  in  cost  of 
maintenance.     Since  the  action  of  heavy  loads  on  an  elastic  rail 
is  to  cause  a  wave  of  translation  in  front  of  each  wheel,  any 
change  in  the  stiffness  or  elasticity  of  the  rail  structure  will 
cause  more  or  less  of  a  shock,  which  must  be  taken  up  and 
resisted  by  the  joint.     The  greater  the  change  in  stiffness  the 
greater  the  shock,  and  the  greater  the  destructive  action  of  the 
shock.     The  perfect  rail-joint  must  keep  both  rail  ends  truly  in 
line  both  laterally  and  vertically,  so  that  the  flange  or  tread  of 
the  wheel  need  not  jump  or  change  its  direction  of  motion  sud- 
denly in  passing  from  one  rail  to  the  other.     A  consideration  of 
all  the  above  requirements  will  show  that  only  a  perfect  welding 
of  rail-ends  would  produce  a  joint  of  uniform  strength  and  stiff- 
ness which  would  give  a  uniform  elastic  wave  ahead  of  each 
wheel.     As  welding  is  impracticable  for  ordinary  railroad  work 
(see   §  230),   some   other   contrivance  is   necessary  which  will 
approach  this  ideal  as  closely  as  may  be. 

238.  Efficiency  of  the  ordinary  angle-bar.     Throughout  the 
middle  portion  of  a  rail  the  rail  acts  as  a  continuous  girder.     If 
we  consider  for  simplicity  that  the  ties  are  unyielding,  the  deflec- 
tion of  suck  a  continuous  girder  between  the  ties  will  be  but 

255 


256  RAILROAD  CONSTRUCTION.  §  239. 

one-fourtli  of  the  deflection  that  would  be  found  if  the  rail  were 
cut  half-way  between  the  ties  and  an  equal  concentrated  load 
were  divided  equally  between  the  two  unconnected  ends.  The 
maximum  stress  for  the  continuous  girder  would  be  but  one-half 
of  that  in  the  cantilevers.  Joining  these  ends  with  rail-joints 
will  give  the  ordinary  "suspended"  joint.  In  order  to  main 
tain  uniform  strength  and  stiffness  the  angle-bars  must  supply 
the  deficiency.  These  theoretical  relations  are  modified  to  an 
unknown  extent  by  the  unknown  and  variable  yielding  of  the  ties. 
From  some  experiments  made  by  the  Association  of  Engineers 
of  Maintenance  of  Way  of  the  P.  R.R.*  the  following  deduc- 
tions were  made : 

1.  The  capacity  of  a  u  suspended  "  joint  is  greater  than  that 
of  a  " supported"  joint — whether  supported  on  one  or  three 
ties.     (See  §  240.) 

2.  That  (with  the  particular  patterns  tested)  the  angle-bars 
alone  can  carry  only  53  to  56$  of  a  concentrated  load  placed 
on  a  joint. 

3.  That  the  capacity  of  the  whole  joint  (angle-bars  and  rail) 
is  only  52.4$  of  the  strength  of  the  unbroken  rail. 

4.  That  the  ineffectiveness  of  the  angle-bar  is  due  chiefly  to 
a  deficiency  in  compressive  resistance. 

Although  it  has  been  universally  recognized  that  the  angle- 
bar  is  not  a  perfect  form  of  joint,  its  simplicity,  cheapness,  and 
reliability  have  caused  its  almost  universal  adoption.  Within  a 
very  few  years  other  forms  (to  be  described  later)  have  been 
adopted  on  trial  sections  and  have  been  more  and  more  extended, 
until  their  present  use  is  very  large.  The  present  time  (1900)  is 
evidently  a  transition  period,  and  it  is  quite  probable  that  within  a 
very  few  years  the  now  common  angle-plate  will  be  as  unknown 
in  standard  practice  as  the  old-fashioned  ' ;  fish-plate  "  is  at  the 
present  time. 

239.  Effect  of  rail  gap  at  joints.  It  has  been  found  that  the 
jar  at  a  joint  is  due  almost  entirely  to  the  deflection  of  the  joint 

*  Headmasters  Association  of  America — Reports  for  1897. 


§  240.  HAIL-FASTENINGS.  257 

and  scarcely  at  all  to  tlie  small  gap  required  for  expansion. 
This  gap  causes  a  drop  equal  to  the  versed  sine  of  the  arc  hav- 
ing a  chord  equal  to  the  gap  and  a  radius  equal  to  the  radius  of 
the  wheel.  Taking  the  extreme  case  (for  a  30-foot  rail)  of  a  f " 
gap  and  a  33"  freight-car  wheel,  the  drop  is  about  roVo""-  ^n 
order  to  test  how  much  the  jarring  at  a  joint  is  due  to  a  gap  be- 
tween the  rails,  the  experiment  was  tried  of  cutting  shallow 
notches  in  the  top  of  an  otherwise  solid  rail  and  running  a  loco- 
motive and  an  inspection  car  over  them.  The  resulting  jarring 
was  practically  imperceptible  and  not  comparable  to  the  jar  pro- 
duced at  joints.  Notwithstanding  this  fact,  many  plans  have 
been  tried  for  avoiding  this  gap.  The  most  of  these  plans  con- 
sist essentially  of  some  form  of  compound  rail,  the  sections 
breaking  joints.  (Of  course  the  design  of  the  compound  rail 
has  also  several  other  objects  in  view.)  In  Fig.  117  are  shown  a 


FIG.  117. — COMPOUND  RAIL  SECTIONS. 

few  of  the  very  many  designs  which  have  been  proposed.  These 
designs  have  invariably  been  abandoned  after  trial.  Another 
plan,  which  has  been  extensively  tried  on  the  Lehigh  Valley 
U.K.,  is  the  use  of  mitered  joints.  The  advantages  gained  by 
their  use  are  as  yet  doubtful,  while  the  added  expense  is  unques- 
tionable. The  "  Roadmasters  Association  of  America"  in  1895 

0 

adopted  a  resolution  recommending  mitered  joints  for  double 
track,  but  their  use  does  not  seem  to  be  growing. 

240.  "Supported,"  "suspended,"  and  "bridge"  joints.  In  a 
supported  joint  the  ends  of  the  rails  are  on  a  tie.  If  the  angle- 
plates  are  short,  the  joint  is  entirely  supported  on  one  tie;  if 
very  long,  it  may  be  possible  to  place  three  ties  under  one  angle- 
bar  and  thus  the  joint  is  virtually  supported  on  three  ties  rather 
than  one.  In  a  suspended  joint  the  ends  of  the  rails  are  midway 
between  two  ties  and  the  joint  is  supported  by  the  two.  There 


258  RAILROAD   CONSTRUCTION.  §  241. 

have  always  been  advocates  of  both  methods,  but  suspended  joints 
are  more  generally  used  than  supported  joints.  The  opponents  of 
three-tie  joints  claim  that  either  the  middle  tie  will  be  too 
strongly  tamped,  thus  making  it  a  supported  joint,  or  that,  if 
the  middle  tie  is  weakest,  the  joint  becomes  a  very  long  (and 
therefore  weak)  suspended  joint  between  the  outer  joint-ties,  or 
that  possibly  one  of  the  outer  joint-ties  gives  way,  thus  breaking 
the  angle-plate  at  the  joint.  Another  objection  which  is  urged 
is  that  unless  the  bars  are  very  long  (say  44  inches,  as  used  on 
the  Mich.  Cent.  E.K.)  the  ties  are  too  close  for  proper  tamp- 
ing. The  best  answer  to  these  objections  is  the  successful  use 
of  these  joints  on  several  heavy-traffic  roads. 

"  Bridge  "-joints  are  similar  to  suspended  joints  in  that  the 
joint  is  supported  on  two  ties,  but  there  is  the  important  differ- 
ence that  the  bridge-joint  supports  the  rail  from  underneath  and 
there  is  no  transverse  stress  in  the  rail,  whereas  the  supported 
joint  requires  the  combined  transverse  strength  of  both  angle- 
bars  and  rail.  A  serious  objection  to  bridge- joints  lies  in  the 
fact  of  their  considerable  thickness  between  the  rail  base  and  the 
tie.  When  joints  are  placed  "  staggered  "  rather  than  u  oppo- 
site "  (as  is  now  the  invariable  standard  practice),  the  ties  sup- 
porting a  bridge-joint  must  either  be  notched  down,  thus 
weakening  the  tie  and  promoting  decay  at  the  cut,  or  else  the 
tie  must  be  laid  on  a  slope  and  the  joint  and  the  opposite  rail 
do  not  get  a  fair  bearing. 

241,  Failures  of  rail-joints.  It  has  been  observed  on  double- 
track  roads  that  the  maximum  rail  wear  occurs  a  few  inches  be- 
yond the  rail  gap  at  the  joint  in  the  direction  of  the  traffic.  On 
single-track  roads  the  maximum  rail  wear  is  found  a  few  inches 
each  side  of  the  joint  rather  than  at  the  extreme  ends  of  the  rail, 
thus  showing  that  the  rail  end  deflects  down  under  the  wheel 
until  (with  fast  trains  especially)  the  wheel  actually  jumps  the 
space  and  strikes  the  rail  a  few  inches  beyond  the  joint,  the 
impact  producing  excessive  wear.  This  action,  which  is  called  the 
"  drop,"  is  apt  to  cause  the  first  tie  beyond  the  joint  to  become 
depressed,  and  unless  this  tie  is  carefully  watched  and  main- 


§242. 


BAIL-FASTENINGS. 


259 


tained  at  its  proper  level,  the  stresses  in  the  angle-bar  may 
actually  become  reversed  and  the  bar  may  break  at  the  top.  The 
angle-bars  of  a  suspended  joint  are  normally  in  compression  at 
the  top.  The  mere  reversal  of  the  stresses  would  cause  the  bars 


FIG.  118. — EFFECT  OF  "WHEEL  DROP"  (EXAGGERATED). 

to  give  way  with  a  less  stress  than  if  the  stress  were  always  the 
same  in  kind.  A  supported  joint,  and  especially  a  three-tie 
joint  (see  §  240),  is  apt  to  be  broken  in  the  same  manner. 

242.  Standard  angle-bars, — An  angle-bar  must  be  so  made 
as  to  closely  fit  the  rails.  The  great  multiplicity  in  the  designs 
of  rails  (referred  to  in  Chapter  IX)  results  in  nearly  as  great 
variety  in  the  detailed  dimensions  of  the  angle-bars.  The  sec- 
tions here  illustrated  must  be  considered  only  as  types  of  the 
variable  forms  necessary  for  each  different  shape  of  rail.  The 
absolutely  essential  features  required  for  a  fit  are  (1)  the  angles 


FIG.  119. — STANDARD  ANGLE-BAR — 80-LB.  RAIL.    M.  C.  R.R. 

of  the  upper  and  lower  surfaces  of  the  bar  where  they  fit  against 
the  rail,  and  (2)  the  height  of  the  bar.     The  bolt-holes  in  the 


260  RAILROAD  CONSTRUCTION.  §  243, 

bar  and  rail  must  also  correspond.  The  holes  in  the  angle-plates 
are  elongated  or  made  oval,  so  that  the  'track- bolts,  which  are 
made  of  corresponding  shape  immediately  under  the  head,  will 
not  be  turned  by  jarring  or  vibration.  The  holes  in  the  rails 
are  made  of  larger  diameter  (by  about  J")  than  the  bolts,  so  as 
to  allow  the  rail  to  expand  with  temperature. 

243.  Later  designs  of  rail-joints.     In  Plate  XVIII  are  shown 
various  designs  which  are  competing  for  adoption.     The  most 
prominent  of  these  (judging  from  the  discussion  in  the  conven- 
tion of  the  Koadrnasters  Association  of  America  in  1897)  are 
the  "  Continuous  "  and  the  "  Weber."     Each  of  them  has  been 
very  extensively  adopted,  and  where  used  are  universally  pre- 
ferred to  angle-plates.      Nearly  all  the  later  designs  embody 
more  or  less  directly  the  principle  of  the  bridge-joint,  i.e.,  sup- 
port the  rail  from  underneath.     An  experience  of  several  years 
will  be  required  to  demonstrate  which  form  of  joint  best  satis- 
fies the  somewhat  opposed  requirements  of  minimum  cost  (both 
initial  and  for  maintenance)  and  minimum  wear  of  rails  and 
rolling  stock. 

TIE-PLATES. 

244.  Advantages,      (a)  As  already  indicated  in  §  204,  the 
life  of  a  soft-wood  tie  is  very  much  reduced  by  "  rail- cutting  " 
and   "  spike-killing,"  such    ties   frequently    requiring   renewal 
long  before  any  serious  decay  has  set  in.     It  has  been  practi- 
cally demonstrated  that  the  "  rail- cutting  "   is  not  due  to  the 
mere  pressure  of  the  rail  on  the  tie,   even  with  a  maximum 
load  on  the   rail,    but   is    due   to    the   impact   resulting   from 
vibration  and  to  the  longitudinal  working  of  the  rail.     It  ha& 
been  proved  that  this  rail-cutting  is  practically  prevented  by 
the  use  of  tie-plates.       (J)  On  curves  there  is  a  tendency  to 
overturn  the  outer  rail  due  to  the  lateral  pressure  on  the  side  of 
the  head.      This  produces  a  concentrated  pressure  of  the  outer 
edge  of  the  base  on  the  tie  which  produces  rail-cutting  and  also 
draws  the  inner  spikes.     Formerly  the  only  method  of  guarding 


PLATE    XVIII. 


SECTION  THROUGH  PLATE  AT  POINT.  SECTION  THROUGH  SPRING-HOUSING. 

RAIL  JOINTS  AND  FROGS. 
(To  face  page  260.) 


§  245.  RAIL-FASTENINGS.  261 

against  this  was  by  the  use  of  "  rail  -braces,"  one    pattern  of 
which  is  shown  in  Fig.  120.     But  it  has  been  found  that  tie- 


FIG.  120. 

plates  serve  the  purpose  even  better,  and  rail-braces  have  been 
abandoned  where  tie-plates  are  used,  (c)  Driving  spikes  through 
holes  in  the  plate  enables  the  spikes  on  each  side  of  the  rail  to 
mutually  support  each  other,  no  matter  in  which  (lateral)  direc- 
tion the  rail  may  tend  to  move,  and  this  probably  accounts  in 
large  measure  for  the  added  stability  obtained  by  the  use  of  tie- 
plates,  (d)  The  wear  in  spikes,  called  ' '  necking, ' '.  caused  by 
the  vertical  vibration  of  the  rail  against  them,  is  very  greatly 
reduced,  (e)  The  cost  is  very  small  compared  with  the  value 
of  the  added  life  of  the  tie,  the  large  reduction  in  the  work  of 
track  maintenance,  and  the  smoother  running  on  the  better  track 
which  is  obtained.  It  has  been  estimated  that  by  the  use  of 
tie-plates  the  life  of  hard-wood  ties  is  increased  from  one  to 
three  years,  and  the  life  of  soft-wood  ties  is  increased  from  three 
to  six  years.  From  the  very  nature  of  the  case,  the  value  of 
tie-plates  is  greater  when  they  are  used  to  protect  soft  ties. 

245,  Elements  of  the  design.  The  earliest  forms  of  tie-plates 
were  flat  on  the  bottom,  but  it  was  soon  found  that  they  would 
work  loose,  allow  sand  and  dirt  to  get  between  the  rail  and  the 
plate  and  also  between  the  plate  and  the  tie,  which  would  cause 
excessive  wear.  Such  plates  are  also  apt  to  produce  an  objec- 
tionable rattle.  Another  fault  of  the  earlier  designs  was  the  use 
of  plates  so  thin  that  they  would  buckle.  The  latest  designs 
have  flanges  or  ' c  teeth ' '  formed  on  the  lower  surface  which 
penetrate  the  tie  about  f"  to  If".  Opinion  is  still  divided  on 
the  question  of  whether  these  teeth  should  run  with  the  grain 


262  RAILROAD  CONSTRUCTION.  §  246. 

or  acro'ss  the  grain.  If  the  flanges  run  with  the  grain,  they 
generally  extend  the  whole  length  of  the  tie-plate — as  in  the 
Wolhaupter  design.  If  the  grain  is  to  be  cut  crosswise,  several 
teeth  about  V  wide  will  be  used — as  in  the  Goldie  design. 


WOLHAUPTER 

^ 

FIG.  121.— TIE-PLATES. 

It  is  a  very  important  feature  that  the  spike-holes  shou/d  be 
so  punched  that  the  spikes  will  fit  closely  to  the  base  of  the  rail. 
Otherwise  a  lateral  motion  of  the  rail  will  be  permitted  which 
will  defeat  one  of  the  main  objects  of  the  use  of  the  plate. 

Another  unsettled  detail  is  the  use  of  ' '  shoulders ' '  on  the 
upper  surface.  On  the  one  hand  it  is  claimed  that  the  use  of 
shoulders  relieves  the  spikes  of  side  pressure  from  the  rail  and 
prevents  "necking."  On  the  other  hand  it  is  claimed  that  if 
the  plain  plate  is  once  properly  set  with  new  spikes  (at  least 
with  spikes  not  already  necked)  the  spikes  will  not  neck  appre- 
ciably, and  that,  as  the  shouldered  plates  cost  more,  the  additional 
expenditure  is  unnecessary. 

The  above  designs  should  be  studied  with  reference  to  the 
manner  in  which  they  fulfill  the  requirements  which  have  been 
already  stated.  As  in  the  case  of  rail-joints,  the  best  forms  of 
tie-plates  are  of  comparatively  recent  design,  and  experience 
with  them  is  still  insufficient  to  determine  beyond  all  question 
which  designs  are  the  best. 

246.  Methods  of  setting.  A  very  important  detail  in  the 
process  of  setting  the  tie-plates  on  the  ties  is  that  the  flanges  or 
teeth  should  penetrate  the  tie  as  far  as  desired  when  the  plates 
are  first  put  in  position.  It  requires  considerable  force  to  press 
the  teeth  into  a  tie.  In  a  few  cases  trackmen  have  depended  on 
the  easy  process  of  waiting  for  passing  trains  to  force  the  teeth 


247. 


RAIL-FASTENINGS. 


263 


down.  Until  the  teeth  are  down  the  spikes  cannot  be  driven 
home,  and  this  apparently  cheap  and  easy  process  results  in  loose 
spikes  and  rails.  If  the  trackmen  neglect  even  temporarily  to 
tighten  these  spikes,  it  will  become  impossible  to  make  them 
tight  ultimately.  The  plates  are  generally  pounded  into  place 
with  a  10-  to  16-pound  sledge-hammer.  A  very  good  method 
was  adopted  once  during  the  construction  of  a  bridge  when  a 
pile-driver  was  at  hand.  The  bridge-ties  were  placed  under  the 
pile-hammer.  The  plates,  accurately  set  to  gauge,  were  then 
forced  in  by  a  blow  from  the  3000-lb.  hammer  falling  2  or  3 
feet. 

SPIKES. 

247.  Requirements.  The  rails  must  be  held  to  the  ties  by  a 
fastening  which  will  not  only  give  sufficient  resistance,  but  which 
will  retain  its  capacity  for  resistance.  It  must  also  be  cheap 
and  easily  applied.  The  ordinary  track-spike  fulfills  the  last 
requirements,  but  has  comparatively  small  resisting  power,  com- 
pared with  screws  or  bolts.  Worse  than  all,  the  tendency  to 
vertical  vibration  in  the  rail  produces  a  series  of  upward  pulls  on 
the  spike  that  soon  loosens  it.  When  motion  has  once  begun 
the  capacity  for  resistance  is  greatly  reduced,  and  but  little  more 
vibration  is  required  to  pull  the  spike  out 
so  much  that  redriving  is  necessary. 
Driving  the  spike  to  place  again  in  the 
same  hole  is  of  small  value  except  as  a 
very  temporary  expedient,  as  its  holding 
power  is  then  very  small.  Redriving  the 
spikes  in  new  holes  very  soon  ' '  spike-kills ' ' 
the  tie.  Many  plans  have  been  devised  to 
increase  the  holding  power  of  spikes,  such 
as  making  them  jagged,  twisting  the  spike, 
swelling  the  spike  at  about  the  center  of  its 
length,  etc.  But  it  has  been  easily  demon- 
strated that  the  fibers  of  the  wood  are  gen-  FlG-  123- 
erally  so  crushed  and  torn  by  driving  such  spikes  that  their 
holding  power  is  less  than  that  of  the  plain  spike. 


264  RAILROAD  CONSTRUCTION.  §  248. 

The  ordinary  spike  (see  Fig.  122)  is  made  with  a  square 
cross-section  which  is  uniform  through  the  middle  of 
its  length,  the  lower  If "  tapering  down  to  a  chisel 
edge,  the  upper  part  swelling  out  to  the  head.  The 
Goldie  spike  (see  Fig.  123)  aims  to  improve  this  form 
by  reducing  to  a  minimum  the  destruction  of  the 
fibers.  To  this  end,  the  sides  are  made  smooth,  the 
edges  are  clean-cut,  and  the  point,  instead  of  being 
chisel-shaped,  is  ground  down  to  a  pyramidal  form. 
Such  fiber-cutting  as  occurs  is  thus  accomplished 
without  much  crushing,  and  the  fibers  are  thus 
pressed  away  from  the  spike  and  slightly  downward. 
Any  tendency  to  draw  the  spike  will  therefore  cause 
FIG.  123.  the  fibers  to  press  still  harder  on  the  spike  and  thus 
increase  the  resistance. 

248.  Driving.     The  holding  power  of  a  spike  depends  largely 
on  how  it  is  driven.     If  the  blows  are  eccentric  and  irregular 
in   direction,  the  hole  will  be  somewhat 

enlarged  and  the  holding  power  largely 
decreased.  The  spikes  on  each  side  of 
the  rail  in  any  one  tie  should  not  be 
directly  opposite,  but  should  be  staggered. 
Placing  them  directly  opposite  will  tend  [ 
to  split  the  tie,  or  at  least  decrease  the 
holding  power  of  the  spikes.  The  direc- 
tion of  staggering  should  be  reversed  in 
the  two  pairs  of  spikes  in  any  one  tie  Fm-  124-  SPIKE-DRIVING. 
(see  Fig.  124).  This  will  tend  to  prevent  any  twisting  of  the  tie 
in  the  ballast,  which  would  otherwise  loosen  the  rail  from  the  tie. 

249.  Screws  and  bolts.     The  use  of  these  abroad  is  very  ex- 
tensive, but  their  use  in  this  country  has  not  passed  the  experi- 
mental stage.     The  screws  are  "  wood  "-screws  (see  Fig.  125), 
having  large   square  heads,  which    are  screwed  down  with  a 
track- wrench.     Holes,  having  the  same  diameter  as  the  base  of 
the  screw-threads,  should  first  be  bored  into  the  tie,  at  exactly 
the  right  position    and  at  the  proper  angle  with  the  vertical. 


249. 


BAIL-FASTENINGS. 


265 


A  light  wooden  frame  is  sometimes  used  to  guide  the  auger  at 
the  proper  angle.  Sometimes  the  large  head  of  the  screw  bears 
directly  against  the  base  of  the  rail,  as  with  the  ordinary 
spike.  Other  designs  employ  a  plate,  made  to  fit  the 
rail  on  one  side,  bearing  on  the  tie  on  the  other  side,  and 
through  which  the  screw  passes.  These  screws  cost 
much  more  than  spikes  and  require  more  work  to  put 
in  place,  but  their  holding  power  is  much  greater 
and  the  work  of  track  maintenance  is  very  much 
less.  Screw-bolts,  passing  entirely  through  the  tie, 
having  the  head  at  the  bottom  of  the  tie  and  the  nut  on  FIG.  125. 
the  upper  side,  are  also  used  abroad.  These  are  quite  difficult 
to  replace,  requiring  that  the  ballast  be  dug  out  beneath  the  tie, 
but  on  the  other  hand  the  occasions  for  replacing  such  a  bolt 
are  comparatively  rare,  as  their  durability  is  very  great.  The 


FIG.  126. 


use  of  screws  or  bolts  increases  the  life  of  the  tie  by  the  avoid- 
ance of  "  spike-killing."  It  is  capable  of  demonstration  that 
the  reduced  cost  of  maintenance  and  the  resulting  improvement 
in  track  would  much  more  than  repay  the  added  cost  of  screws 
and  bolts,  but  it  seems  impossible  to  induce  railroad  directors  to 
authorize  a  large  and  immediate  additional  expenditure  to  make 
an  annual  saving  whose  value,  although  unquestionably  consider- 
able, cannot  be  exactly  computed. 


266  RAILROAD  CONSTRUCTION.  §  250. 

250.  "Wooden  spikes."     Among  the  regulations  for  track- 
laying  given  in  §  208,  mention  was  made  of  wooden  "spikes," 
or  plugs,  which  are  used  to  fill  up   the   holes  when   spikes  are 
withdrawn.     The  value  of  the  policy  of  filling  up  these  holes  is 
unquestionable,  since  the  expense  is  insignificant  compared  with 
the  loss  due  to  the  quick  and  certain  decay  of  the  tie  if  these 
holes  are  allowed  to  fill  with  water  and  remain  so.     But  the 
method  of  making  these  plugs  is  variable.      On  some  roads  they 
are   ' '  hand-made ' ?    by  the  trackmen  out  of  otherwise  useless 

scraps    of    lumber,    the    work    being    done    at    odd 
moments.      This  policy,   while    apparently  cheap,    is 
not  necessarily  so,  for  the   hand-made  plugs  are  ir- 
regular in  size  and  therefore  more  or  less  inefficient. 
It    is  also    quite    probable    that  if    a   track    gang   is 
required  to  make  their  own   plugs,    they  may  spend 
time    on    these   very  cheap    articles    which    could   be 
more  profitably  employed  otherwise.      Since  the  holes 
made  by  the  spikes  are  larger  at  the  top  than  they  are 
near  the  bottom,  the  plugs  should  not  be  of  uniform 
cross  section    but    should    be    slightly   wedge-shaped. 
The  "Goldie  tie-plug"   (see  Fig.  127)  has    been  de- 
VJ5°o      signed   to    fill  these   requirements.      Being   machine- 
made,  they  are  uniform  in  size ;   they  are  of  a  shape 
which  will  best  fit  the  hole ;  they  can  be  furnished  of  any  desired 
wood,  and  at  a  cost  which  makes  it  a  wasteful  economy  to  at- 
tempt to  cut  them  by  hand. 

TEACK- BOLTS    AND  NTJT- LOCKS. 

251.  Essential  requirements.       The  track-bolts   must   have 
sufficient  strength  and  must  be  screwed  up  tight  enough  to  hold 
the  angle-plates  against  the  rail  with  sufficient  force  to  develop 
the  full  transverse  strength  of  the  angle-bars.      On  the  other 
hand  the  bolts  should  not  be  screwed  so  tight  that  slipping  may 
not  take  place  when  the  rail  expands  or  contracts  with  temperature. 
It  would  be  impossible  to  screw  the  bolts  tight  enough  to  prevent 


§  252.  KAIL-FASTENINGS.  267 

slipping  during  the  contraction  due  to  a  considerable  fall  of 
temperature  on  a.  straight  track,  but  when  the  track  is  curved, 
or  when  expansion  takes  place,  it  is  conceivable  that  the  resist- 
ance of  the  ties  in  the  ballast  to  lateral  motion  may  be  less  than 
the  resistance  at  the  joint.  A  test  to  determine  this  resistance 
was  made  by  Mr.  A.  Torrey,  chief  engineer  of  the  Mich.  Cent. 
K.R.,  using  SO-lb.  rails  and  ordinary  angle-bars,  the  bolts  being 
screwed  up  as  usual.  It  required  a  force  of  about  31000  to 
35000  Ibs.  to  start  the  joint,  which  would  be  equivalent  to  the 
stress  induced  by  a  change  of  temperature  of  about  22°.  But 
if  the  central  angle  of  any  given  curve  is  small,  a  comparatively 
small  lateral  component  will  be  sufficient  to  resist  a  compression 
of  even  35000  Ibs.  in  the  rails.  Therefore  there  will  ordinarily 
be  no  trouble  about  having  the  joints  screwed  too  tight.  The 
vibration  caused  by  the  passage  of  a  train  reduces  the  resistance 
to  slipping.  This  vibration  also  facilitates  an  objectionable 
feature,  viz.,  loosening  of  the  nuts  of  the  track-bolts.  The  bolt 
is  readily  prevented  from  turning  by  giving  it  a  form  which  is 
not  circular  immediately  under  the  head  and  making  corre- 
sponding holes  in  the  angle- plate.  Square  holes  would  answer 
the  purpose,  except  that  the  square  corners  in  the  holes  in  the 
angle-plates  would  increase  the  danger  of  fracture  of  the  plates. 
Therefore  the  holes  (and  also  the  bolts,  under  the  head)  are 
made  of  an  oval  form,  or  perhaps  a  square  form  with  rounded 
corners,  avoiding  angles  in.  the  outline. 

The  nut-locks  should  be  simple  and  cheap,  should  have  a  life 
at  least  as  long  as  the  bolt,  should  be  effective,  and  should  not 
lose  their  effectiveness  with  age.  Many  of  the  designs  that 
have  been  tried  have  been  failures  in  one  or  more  of  these 
particulars,  as  will  be  described  in  detail  below. 

252.  Design  of  track-bolts.  In  Fig.  128  is  shown  a  common 
design  of  track-bolt.  In  its  general  form  this  represents 
die  bolt  used  on  nearly  all  roads,  being  used  not  only 
with  the  common  angle-plates,  but  also  with  many  of  the  im- 
proved designs  of  rail-joints.  The  variations  are  chiefly  a 
general  increase  in  size  to  correspond  with  the  increased 


268 


RAILROAD  CONSTRUCTION. 


§253. 


FIG.  128  —  TRACK-BOLT. 


weight  of  rails,  besides  variations  in    detail  dimensions  which 
are    frequently   unimportant.       The  diameter  is  usually  £"  to 

•J" ;  V  bolts  are  sometimes  used  for 
the  heaviest  sections  of  rails.  As 
to  length,  the  bolts  should  not  ex- 
tend more  than  -J-"  outside  of  the 
nut  when  it  is  screwed  up.  If  it 
extends  farther  than  this,  it  is  liable 
to  be  broken  off  by  a  possible  derail- 
ment at  that  point.  The  lengths  used 
vary  from  3J",  which  may  be  used 
with  60  Ibs.  rails,  to  5",  which  is 
required  with  100-lb.  rails.  The 
length  required  depends  somewhat  on 
the  type  of  nut-lock  used. 
253,  Design  of  nut-locks.  The  designs  for  nut- locks  may  be 
divided  into  three  classes :  (a)  those  depending  entirely  on  an 
elastic  washer  which  absorbs  the  vibration  which  might  other- 
wise induce  turning ;  (b)  those  which  jam  the  threads  of  the 
bolt  and  nut  so  that,  when  screwed  up,  the  frictional  resistance 
is  too  great  to  be  overcome  by  vibration  ;  (c)  the  ' '  positive  ' ' 
nut-locks — those  which  mechanically  hold  the  nut  from  turning. 
Some  of  the  designs  combine  these  principles  to  some  extent. 
The  "  vulcanized  fiber"  nut-lock  is  an  example  of  the  first 
class.  It  consists  essentially  of  a  rubber  washer  which  is  pro- 
tected by  an  iron  ring.  When  first  placed  this  lock  is  effective, 
but  the  rubber  soon  hardens  and  loses  its  elasticity  and  it  is  then 
ineffective  and  worthless.  Another  illustration  of  class  (a)  is 
the  use  of  wooden  blocks,  generally  of  V  to  2''  oak,  which 
extend  the  entire  length  of  the  angle- bar,  a  single  piece  forming 
the  washer  for  the  four  or  six  bolts  of  a  joint.  This  form  is 
cheap,  but  the  wood  soon  shrinks,  loses  its  elasticity,  or  decays  so 
that  it  soon  becomes  worthless,  and  it  requires  constant  adjust- 
ment to  keep  it  in  even  tolerable  condition.  The  "Yerona" 
nut-lock  is  another  illustration  of  class  (a)  which  also  combines 
some  of  the  positive  elements  of  class  (c).  It  is  made  of 


§  253. 


RAIL-  FASTENINGS. 


269 


tempered  steel  and,  as  shown  in  Fig.  129,  is  warped  and  has 
sharp  edges  or  points.  The  warped  form  furnishes  the  element 
of  elastic  pressure  when  the  nut  is  screwed  up.  The  steel 
being  harder  than  the  iron  of  the  angle-bar  or  of  the  nut,  it 
bites  into  them,  owing  to  the  great  pressure  that  must  exist 


NATIONAL 


JONES 

EXCELSIOR-, 

FIG.  129. — TYPES  OF  NUT-LOCKS. 


when  the  washer  is  squeezed  nearly  flat,  and  thus  prevents  any 
backward  movement,  although  forward  movement  (or  tighten- 
ing the  bolt)  is  not  interfered  with.  The  ' '  National ' '  nut-lock 
is  a  type  of  the  second  class  (J),  in  which,  like  the  "  Harvey  " 
nut-lock,  the  nut  and  lock  are  combined  in  one  piece.  With 
six-bolt  angle-bars  and  30-foot  rails,  this  means  a  saving  of  2112 
pieces  on  each  mile  of  single  track.  The  "  National  "  nuts  are 
open  on  one  side.  The  hole  is  drilled  and  the  thread  is  cut 
slightly  smaller  than  the  bolt,  so  that  when  the  nut  is  screwed 


270  RAILROAD    CONSTRUCTION.  §  253. 

up  it  is  forced  slightly  open  and  therefore  presses  on  the  threads 
of  the  bolt  with  such  force  that  vibration  cannot  jar  it  loose. 
Unlike  the  "  National"  nut,  the  "  Harvey"  nut  is  solid,  but 
the  form  of  the  thread  is  progressively  varied  so  that  the  thread 
pinches  the  thread  of  the  bolt  and  the  frictional  resistance  to- 
turning  is  too  great  to  be  affected  by  vibration. 

The  u  Jones"  nut-lock,  belonging  to  class  (<?),  is  a  type  of 
a  nut-lock  that  does  not  depend  on  elasticity  or  jamming  of 
screw-threads.  It  is  made  of  a  thin  flexible  plate,  the  square 
part  of  which  is  so  large  that  it  will  not  turn  after  being  placed 
on  the  bolt.  After  the  nut  is  screwed  up,  the  thin  plate  is  bent 
over  so  that  the  re-entrant  angle  of  the  plate  engages  the  corner 
of  the  nut  and  thus  mechanically  prevents  any  turning.  The 
metal  is  supposed  to  be  sufficiently  tough  to  endure  without 
fracture  as  many  bendings  of  the  plate  as  will  ever  be  desired. 
Nut-locks  of  class  (c)  are  not  in  common  use. 


CHAPTEK  XI. 
SWITCHES  AND  CROSSINGS. 

SWITCH    CONSTRUCTION. 

254,  Essential  elements  of  a  switch.  Flanges  of  some  sort  are 
a  necessity  to  prevent  car-wheels  from  running  off  from  the  rails 
on  which  they  may  be  moving.  But  the  flanges,  although  a 
necessity,  are  also  a  source  of  complication  in  that  they  require 
some  special  mechanism  which  will,  when  desired,  guide  the 
wheels  out  from  the  controlling  influence  of  the  main-line  rails. 
This  must  either  be  done  by  raising  the  wheels  high  enough 
so  that  the  flanges  may  pass  over  the  rails,  or  by  breaking  the 
continuity  of  the  rails  in  such  a  way  that  channels  or  ' ;  flange 
spaces  "  are  formed  through  the  rails.  An  ordinary  stub  switch 
breaks  the  continuity  of  the  main-line  rails  in  three  places,  two 
of  them  at  the  switch- block  and  one  at  the  frog.  The  Wharton 
switch  avoids  two  of  these  breaks  by  so  placing  inclined  planes 
that  the  wheels,  rolling  on  their  flanges,  will  surmount  these 
inclines  until  they  are  a  little  higher  than  the  rails.  Then  the 
wheels  on  the  side  toward  which  the  switch  runs  are  guided 
over  and  across  the  main  rail  on  that  side.  This  rise  being  ac- 
complished in  a  short  distance,  it  becomes  impracticable  to 
operate  these  switches  except  at  slow  speeds,  as  any  sudden 
change  in  the  path  of  the  center  of  gravity  of  a  car  causes  very 
destructive  jars  both  to  the  switch  and  to  the  rolling  stock.  The 
other  general  method  makes  a  break  in  one  main  rail  (or  both) 
at  the  switch-block.  In  both  methods  the  wheels  are  led  to  one 
side  by  means  of  the  "lead  rails,"  and  finally  one  line  of  wheels 
passes  through  the  main  rail  on  that  side  by  means  of  a  "  frog. ' y 
There  are  some  designs  by  which  even  this  break  in  the  main 
rail  is  avoided,  the  wheels  being  led  over  the  main  rail  by  means- 

271 


272  RAILROAD   CONSTRUCTION.  §  255. 

of  a  short  movable  rail  which  is  on  occasion  placed  across  the 
main  rail,  but  such  designs  have  not  come  into  general  use. 

255.  Frogs.  Frogs  are  provided  with  two  channel-ways  or 
•"  flange  spaces  "  through  which  the  flanges  of  the  wheels  move. 
Each  channel  cuts  out  a  parallelogram  from  the  tread  area. 
Since  the  wheel-tread  is  always  wider  than  the  rail,  the  wing 
rails  will  support  the  wheel  not  only  across  the  space  cut  out  by 


I 

FIG.  130. — DIAGRAMMATIC  DESIGN  OF  FROG. 

the  channel,  but  also  until  the  tread  has  passed  the  point  of  the 
frog  and  can  obtain  a  broad  area  of  contact  on  the  tongue  of  the 
frog.  This  is  the  theoretical  idea,  but  it  is  very  imperfectly 
realized.  The  wing  rails  are  sometimes  subjected  to  excessive 
wear  owing  to  ' '  hollow  treads  ' '  on  the  wheels — owing  also  to 
the  frog  being  so  flexible  that  the  point  "ducks"  when  the 
wheel  approaches  it.  On  the  other  hand  the  sharp  point  of  the 
frog  will  sometimes  cause  destructive  wear  on  the  tread  of  the 
wheel.  Therefore  the  tongue  of  the  frog  is  not  carried  out  to 
the  sharp  theoretical  point,  but  is  purposely  somewhat  blunted. 
But  the  break  which  these  channels  make  in  the  continuity  of 
the  tread  area  becomes  extremely  objectionable  at  high  speeds, 
Jbeing  mutually  destructive  to  the  rolling  stock  and  to  the  frog. 
The  jarring  has  been  materially  reduced  by  the  device  of 
"spring  frogs"  —to  be  described  later.  Frogs  were  originally 
made  of  cast  iron — then  of  cast  iron  with  wearing  parts  of  cast 
steel,  which  were  fitted  into  suitable  notches  in  the  cast  iron. 
This  form  proved  extremely  heavy  and  devoid  of  that  elasticity 
of  track  which  is  necessary  for  the  safety  of  rolling  stock 
and  track  at  high  speeds.  The  present  universal  practice  is  to 
build  the  frog  up  of  pieces  of  rails  which  are  cut  or  bent  as  re- 
quired. These  pieces  of  rails  (at  least  -four)  are  sometimes 


§  256.  SWITCHES  AND  CROSSINGS  273 

assembled  by  riveting  them  to  a  flat  plate,  but  this  method  is 
now  but  little  used,  except  for  very  light  work.  The  usual 
practice  is  now  chiefly  divided  between  "  bolted  "  and  "  keyed  " 
frogs.  In  each,  case  the  space  between  the  rails,  except  a  suffi- 
cient flange- way,  is  filled  with  a  cast-iron  filler  and  the  whole 
assemblage  of  parts  is  suitably  bolted  or  clamped  together,  as  is 
illustrated  in  Plate  XVIII.  The  operation  of  a  spring-rail  frog 
is  evident  from  the  figure.  Since  a  siding  is  usually  operated  at 
slow  speed,  while  the  main  track  may  be  operated  at  fast  speed, 
a  spring-rail  frog  will  be  so  set  that  the  tread  is  continuous  for 
the  main  track  and  broken  for  the  siding.  This  also  means  that 
the  spring  rail  will  only  be  moved  by  trains  moving  at  a  (pre- 
sumably) slow  speed  on  to  the  siding.  For  the  fast  trains  on  the 
main  line  such  a  frog  is  substantially  a  "fixed  "  frog  and  has  a 
tread  which  is  practically  continuous. 

256.  To  find  the  frog  number.     The  frog  number  (n)  equals 
the  ratio  of  the  distance  of  any  point  on  the  tongue  of  the  frog 
from  the  theoretical  point  of  the  frog  divided  by  the  width  of 
the  tongue  at  that  point,    i.e.  —  lie  -=-  ab  (Fig.    130).      This 
value  may  be  directly  measured  by  applying  any  convenient 
unit   of  measure   (even  a  knife,  a  short  pencil,  etc.)  to  some 
point  of  the  tongue  where  the  width  just  equals  the  unit  of 
measure,  and  then  noting  how  many  times  the  unit  of  measure 

is  contained  in  the  distance  from   that  place  to  the  theoretical 

«» 

point.  But  since  <?,  the  theoretical  point,  is  not  so  readily 
determinate  with  exactitude,  it  being  the  imaginary  inter- 
section of  the  gauge  lines,  it  may  be  more  accurate  to  measure 
de,  ab,  and  Jis ;  then  n,  the  frog  number,  =4  hs  -=-  (ab  +  de). 
If  the  frog  angle  be  called  F,  then 

n  =  he  -T-  ab  =  hs  -r-  (ab  +  de)  =  •£  cot  \F  ; 
i,e..  cot  \F  =  2n. 

257.  Stub  switches.     The  use  of  these,  although  once  nearly 
universal,    has  been    practically  abandoned    as    turnouts    from 
main  track  except  for  the  poorest  and  cheapest  roads.     In  some 
States,   their  use  on  main  traek  is  prohibited  by  law.     They 


274 


RAILROAD  CONSTRUCTION. 


257. 


have  the  sole  merit  of  cheapness  with  adaptability  to  the  cir- 
cumstances of  very  light  traffic  operated  at  slow  speed  when  a 
considerable  element  of  danger  may  be  tolerated  for  the  sake  of 
economy.  The  rails  from  A  to  £  (see  Fig.  131*)  are  not  fastened 


FIG.  131. — STUB  SWITCH. 

to  the  ties ;  they  are  fastened  to  each  other  by  tie-rods  which 
keep  them  at  the  proper  gauge ;  at  and  back  of  B  they  are 
securely  spiked  to  the  ties,  and  at  A  they  are  kept  in  place  by 
the  connecting  bar  (C)  fastened  to  the  switch-stand.  One  great 
objection  to  the  switch  is  that,  in  its  usual  form,  when  operated 
as  a  trailing  switch,  a  derailment  is  inevitable  if  the  switch  is- 
misplaced.  The  very  least  damage  resulting  from  such  a  derail- 
ment must  include  the  bending  or  breaking  of  the  tie-rods  of  the 
switch -rail.  Several  devices  have  been  invented  to  obviate  tins- 
objection,  some  of  which  succeed  very  well  mechanically, 
although  their  added  cost  precludes  any  economy  in  the  total 
cost  of  the  switch.  Another  objection  to  the  switch  is  the 
looseness  of  construction  which  makes  the  switches  objectionable 
at  high  speeds.  The  gap  of  the  rails  at  the  head-block  is  always 
considerable,  and  is  sometimes  as  much  as  two  inches.  A 

*  The  student  should  at  once  appreciate  that  in  Fig.  131,  as  well  as  in 
nearly  all  the  remaining  figures  in  this  chapter, 'it  becomes  necessary  to  use 
excessively  large  frog  angles,  short  radii,  and  a  very  wide  gauge  in  order  to 
illustrate  the  desired  principles  with  figures  which  are  sufficiently  small  for 
the  page.  In  fact,  the  proportions  used  in  the  figures  are  such  that  serious 
mechanical  difficulties  would  be  encountered  if  they  were  used.  These 
difficulties  are  here  ignored  because  they  can  be  neglected  in  the  proportions 
used  in  practice. 


SWITCHES  AND  CROSSINGS. 


275 


driving-wheel  with  a  load  of  12000  to  20000  pounds,  jumping 
this  gap  with  any  considerable  velocity,  will  do  immense  damage 
to  the  farther  rail  end,  besides  producing  such  a  stress  in  the 
construction  that  a  breakage  is  rendered  quite  likely,  and  such  a 
breakage  might  have  very  serious  consequences. 

258.  Point  switches.  The  essential  principle  of  a  point 
switch  is  illustrated  in  Fig.  132.  As  is  shown,  one  main  rail 
and  also  one  of  the  switch-rails  is  unbroken  and  immovable. 


FIG.  132.— POINT  SWITCH. 

The  other  main  rail  (from  A  to  F)  and  the  corresponding 
portion  of  the  other  lead  rail  are  substantially  the  same  as  in  a 
stub  switch.  A  portion  of  the  main  rail  (AB)  and  an  equal 
length  of  the  opposite  lead  rail  (usually  15  to  24  feet  long)  are 
fastened  together  by  tie-rods.  The  end  at  A  is  jointed  as  usual 
and  the  other  end  is  pointed,  both  sides  being  trimmed  down 
so  that  the  feather  edge  at  B  includes  the  web  of  the  rail.  In 
order  to  retain  in  it  as  much  strength  as  pos- 
sible, the  point-rail  is  raised  so  that  it  rests 
on  the  base  of  the  stock- rail,  one  side  of  the 
l)ase  of  the  point-rail  being  entirely  cut  away. 
As  may  be  seen  in  Fig.  133,  although  the 
influence  of  the  point  of  the  rail  in  moving 
the  wheel-flange  away  from  the  stock-rail  is 
really  zero  at  that  point,  yet  the  rail  has  all 
the  strength  of  the  web  and  about  one -half 

"EVn    1 QQ 

that   of   the  base — a   very   fair    angle-iron. 

The  planing  runs  back  in  straight  lines,  until  at  about  six  or 

seven  feet  back  from   the  point  the  full  width  of  the  head  is 


276 


EAILEOAD  CONSTRUCTION. 


§259. 


obtained.     The  full  width  of  the  base  will  only  be  obtained   at 
about    13    feet   from  the   point.      An    80-lb.  rail  is  5    inches 


FIG.  134.— GROUND  LEVER  FOB  THROWING  A  SWITCH. 

wide  at  the  base.  Allowing  f"  more  for  a  spike  between 
the  rails,  this  gives  5£"  as  the  minimum  width  between  rail 
centers  at  the  joint.  The  minimum  angle  of 
the  switch-point  (using  a  15-foot  point  rail) 
is  therefore  the  angle  whose  tangent  is 

5  75 
— - - — —~  =  .03914,  which  is  the  tangent  of 

1°  50'.  Switch-rails  are  sometimes  used  with 
a  length  of  24  feet,  which  reduces  the  angle 
of  the  switch- point  to  1°  09'. 


259.  Switch-stands,  The  simplest  and 
cheapest  form  is  the  "  ground  lever,"  which 
has  no  target.  The  radius  of  the  circle  de- 
scribed by  the  connecting-rod  pin  is  precisely 
one-half  the  throw.  From  the  nature  of  the 
motion  the  device  is  practically  self-locking  in 
either  position,  padlocks  being  only  used  to 
prevent  malicious  tampering.  The  numerous 
designs  of  upright  stands  are  always  combined 
with  targets,  one  design  of  which  is  illustrated 
in  Fig.  135.  When  the  road  is  equipped 
with  interlocking  signals,  the  switch-throw 
mechanism  forms  a  part  of  the  design. 
FlG-  135-  260.  Tie-rods.  These  are  fastened  to  the 

webs  of  the  rails  by  means  of  lugs  which  are  bolted  on,  there 


§  261.  SWITCHES  AND  CROSSINGS.  277 

being  usually  a  hinge- joint  between  the  rod  and  the  lug.  Four 
such  tie-rods  are  generally  necessary.  The  first  rod  is  some- 
tunes  made  without  hinges,  which  gives  additional  stiffness  to 
the  comparatively  weak  rail-points.  The  old  fashioned  tie-rod, 
having  jaws  fitting  the  base  of  the  rail,  was  almost  universally 
used  in  the  days  of  stub  switches.  One  great  inconvenience 
in  their  use  lies  in  the  fact  that  they  must  be  slipped  on,  one  by 
one,  over  the  free  ends  of  the  switch -rails.  Sometimes  the 
lugs  are  fastened  to  the  rail-webs  by  rivets  instead  of  bolts. 


FIG.  136.— Fouiis  OF  TIE-RODS. 

261.  Guard-rails.  As  shown  in  Figs.  131  and  132,  guard- 
rails are  used  on  both  the  main  and  switch  tracks  opposite  the 
frog-point.  Their  function  is  not  only  to  prevent  the  possibil- 
ity of  the  wheel-flanges  passing  on  the  wrong  side  of  the  frog- 
point,  but  also  to  save  the  side  of  the  frog-tongue  from  exces- 
sive wear.  The  necessity  for  their  use  may  be  realized  by 
noting  the  very  apparent  wear  usually  found  on  the  side  of  the 
head  of  the  guard-rail.  The  flange- way  space  between  the 
heads  of  the  guard-rail  and  wheel-rail  therefore  becomes  a 
definite  quantity  and  should  equal  about  two  inches.  Since  this 
is  less  than  the  space  between  the  heads  of  ordinary  (say 
80-pound)  rails  when  placed  base  to  base,  to  say  nothing  of  the 
f "  necessary  for  spikes,  it  becomes  necessary  to  cut  the  flange 
of  the  guard-rail.  The  length  of  the  rail  is  made  from  10  to 
15  feet,  the  ends  being  bent  as  shown  in  Fig.  132,  so  as  to 


278 


RAILROAD   CONSTRUCTION. 


§ 


prevent  the  possibility  of  the  end  of  the  rail  being  struck  by  a 
wheel-flange. 

MATHEMATICAL    DESIGN    OF    SWITCHES. 

In  all  of  the  following  demonstrations  regarding  switches, 
turnouts,  and  crossovers,  the  lines  are  assumed  to  represent  the 
gauge-lines — i.e.,  the  lines  of  the  inside  of  the  head  of  the 
rails. 

262.  Design  with  circular  lead-rails.  The  simplest  method 
is  to  consider  that  the  lead-rails  curve 
out  from  the  main  track-rails  by  arcs 
of  circles  which  are  tangent  to  the  main 
rails  and  which  extend  to  the  frog-point 
F.  The  simple  curve  from  D  to  F  is 
of  such  radius  that  (r  +  J^)  vers  F  =  gy 
in  which  F  =  the  frog  angle,  g  = 


FIG.  137. 


gauge,    Z  =  the 
r  —  the   radius 
switch -rails. 


"lead"   (BF),  and 
of   me  center  of  the 


9 


Also      BF-=r  BD  —  cot 


=         BF-L. 


Also 


L  =  g  cot 
£=(>•  +  iff)  sin 


(74> 


(75) 
(76). 
(77). 


These  formulae  involve  the  angle  F.  As  shown  in  Table  III, 
the  angles  (F)  are  always  odd  quantities,  and  their  trigonometric 
functions  are  somewhat  troublesome  to  obtain  closely  with 
ordinary  tables.  The  formulae  may  be  simplified  by  substitut- 
ing the  frog-number  n,  from  the  relation  that  n  =  J  cot 
Since 

r  —  \g  =  L  cot  F  and  r  +  \g  ==  Z  cosec  F, 


§  262.  SWITCHES  AND  CROSSINGS.  279 

then  r  =  \L  (cot  F '-f  cosec  F) 

=  \g  cot  \F  (cot  F-\-  cosec  ^) 

=  ^  cot2  \F,  since  (cot  a  +  cosec  a)  =  cot  Ja 

=  2yw8.      .     '.     .     >     ....     .      •     (78) 

Also  Z  =  20w,      .     .     .     .     |i   .    V    ,     .     .     (79). 

from  which       r  =  n  ^  L.       .     ..     .     .     .     .     .     •     (80) 

These  extremely  simple  relations  may  obviate  altogether  the 
necessity  for  tables,  since  they  involve  only  the  frog-number  and 
the  gauge.  On  account  of  the  great  simplicity  of  these  rules, 
they  are  frequently  used  as  they  are,  regardless  of  the  fact  that 
the  curve  is  never  a  uniform  simple  curve  from  switch-block  to 
frog.  In  the  first  place  there  is  a  considerable  length  of  the 
gauge-line  within  the  frog,  which  is  straight  unless  it  is  pur- 
posely curved  to  the  proper  curve  while  being  manufactured, 
which  is  seldom  if  ever  done — except  for  the  very  large-angled 
frogs  used  for  street-railway  work,  etc.  It  is  also  doubtful 
whether  the  switch-rails  (BA,  Fig.  131)  are  bent  to  the  com- 
puted curve  when  the  rails  are  set  for  the  switch.  The  switch- 
rails  of  point  switches  are  straight,  thus  introducing  a  stretch  of 
straight  track  which  is  about  one-fifth  of  the  total  length  of  the 
lead-rails.  The  effect  of  these  modifications  on  the  length  and 
radius  of  the  lead-rails  will  be  developed  and  discussed  in  the 
next  four  sections. 

The  throw  (t)  of  a  stub  switch  depends  on  the  weight  of  the 
rail,  or  rather  on  the  width  of  its  base.  The  throw  must  be  at 
least  f"  more  than  that  width.  The  head-block  should  there- 
fore be  placed  at  such  a  distance  from  the  heel  of  the  switch  (B) 
that  the  versed  sine  of  the  arc  equals  the  throw.  These  points 
must  be  opposite  on  the  two  rails,  but  the  points  on  the  two  rails 
where  these  relations  are  exactly  true  will  not  be  opposite. 
Therefore,  instead  of  considering  either  of  the  two  radii  (r  +  %g) 
and  (r  —  %g\  the  mean  radius  r  is  used.  Then  (see  Fig.  137) 

vers  KOQ  =  t  -7-  r, 


280 


RAILROAD  CONSTRUCTION. 


§263. 


and  the  length  of  the  switch- rails  is 

=  rsmJWQ. 


(81) 


These  relations  develop  another  disadvantage  in  the  use  of  a 
stub  switch.  The  required  value  of  BG,  using  a  No.  10  frog 
and  80-pound  rail,  is  30.1  feet  —  slightly  more  than  a  full  rail 
length.  It  would  be  unsafe  to  leave  so  much  of  the  track  un- 
spiked  from  the  ties.  Whether  this  is  obviated  by  spiking  down 
a  portion  of  the  switch-rails  (virtually  shortening  the  lead)  or  by 
moving  the  switch-block  nearer  the  heel  of  the  switch  (shorten- 
ing the  switch-rails),  but  still  maintaining  the  required  throw, 
the  theoretical  accuracy  of  the  curve  is  hopelessly  lost. 

263.  Effect  of  straight  frog-rails,  A  portion  of  the  ends  of 
the  rails  of  a  frog  are  free  and  may  be  bent  to  conform  to  the 

switch-rail  curve,  but  there  is  a  con- 
siderable portion  which  is  fitted  to  the 
cast-iron  filler,  and  this  portion  is  always 
straight.  Call  the  length  of  this  straight 
portion  back  from  the  frog-point  f 
(=  FH,  Fig.  138).  Then  we  have  * 

f  +  iff  =  (9  ~ 


sn 


-*-  vers 


vers 


FIG.  138. 


-v-  2>. 

vers  F 

»J? 


(82) 


=  L  =  (g  -/sin  F)  cot 

—  %gn  -/sin  F  cot  \F-\-f  cos  F 
=  2gn  -/(I  +  cos  F)  +/  cos  F 


(83) 


Since  r  —  \g  =  (Z  —  /  sec  F)  cot  F,  and 
r  +  \g  =  (L  —f  cos  F)  cosec  F, 


§  264.  SWITCHES  AND  CROSSINGS  281 

r  =  %L  (cot  F-\-  cosec  F)  -  kf  sec  F  cot  F—  \f  cos  F  cosec  F 


r  =  Ln  —  \f  cc 
=  Ln  —fn.     Then  from  (83) 

r  =  Zyri*  —  I2fn.     .  "  .     .     .     ...     (84) 


264.  Effect  of  straight  point-rails.  The  "  point  switches," 
now  so  generally  used,  have  straight  switch-rails.  This  requires 
an  angle  in  the  alignment  rather  than  turning  off  by  a  tangential 
curve.  The  angle  is,  however,  very  small  (between  1°  and  2°), 
and  the  disadvantages  of  this  angle  are  small  compared  with  the 
very  great  advantages  of  the  device. 


x 


MN  =  fc 


XW 


JZ.  1  

f—                                M 

0 

1 

i"            ; 

•"pf" 

FIG.  139. 

FM 

g  —  * 

T^f 


2  sin  i(F  -  a) 

^-'i 


2  sin  %(F  +  a)  sin  J(F  —  a) 

g  —  k 
cos  a  —  cos  F' 


(85) 


282 


RAILROAD  CONSTRUCTION. 


§  265. 


BF  =  L  =  FM  cos 

=  (g-  k)  cot 


a)  +  DN 
a)  +  DN. 


(86) 


265,  Combined  effect  of  straight  frog-rails  and  straight  point- 
rails.  It  becomes  necessary  in  this  case  to  find  a  curve  which 
shall  be  tangent  to  both  the  point-rail  and  the  frog-rail.  The 
curve  therefore  begins  at  JJ/,  its  tangent  making  an  angle  of  OL 
(usually  1°  50')  with  the  main  rail,  and  runs  to  H.  The  central 


FIG.  140. 


angle  of  the  curve  is  therefore  (F  —  a).     The  angle  of  the  chord 
HM  with  the  main  rails  is  therefore 


~*g  - 


HM 


-  a) 


2  sin  %(F  +  a]  sin  %(F  —  a) 

__  g  -/sin  F  —  km 
cos  a  —  cos  F 

8T  =  2r  sin  %(F  -  a).    '. 


(87) 

(88) 


§266. 


SWITCHES  AND  CROSSINGS. 


283 


BF=L  =  JIMcos  t(F+  «)  +/CGS  F+  DN 

=  (g-f^F-  &)  cot  i(F+  a)  +  /  cos  F+  DN.  (89) 


It  may  be  more  simple,  if  (r  +  \g)  has  already  been  com- 
puted, to  write 


20 
(/• 


sn 


—  a  cos 


a)  +/cos 

in  F  —  sin  a)  +  /  cos  ^  + 


.     (90) 


266.  Comparison  of  the  above  methods.  Computing  values 
for  r  and  L  by  the  various  methods,  on  the  uniform  basis  of  a 
Ko.  9  frog,  standard  gauge  4'  8f  ,/  =  3'.  37,  &  =  5f"=  0'.479, 
])N  =15'  0",  and  a:  =  1°  50',  we  may  tabulate  the  compara- 
tive results: 


§  262. 
Simple  circle 
Curved  frog  r. 
Curved  switch-r. 

§263. 
Straight  frog-r. 
Curved  switch-r. 

§264. 
Curved  frog-r. 
Straight  switch-r. 

§265. 
Straight  f  rog  r. 
Straight  switch  r. 

r 

762.75 

702.00 

747.48 

681.16 

Deg.  of  curve 

7°  31' 

8°  IV 

7°  40' 

8°  25' 

L 

84.75 

81.37 

74.00 

72.13 

This  shows  that  the  effect  of  using  straight  frog-rails  and 
straight  switch-rails  is  to  sharpen  the  curve  and  shorten  the  lead 
in  each  case  separately,  and  that  the  combined  effect  is  still 
greater.  The  effect  of  the  straight  switch -rails  is  especially 
marked  in  reducing  the  length  of  lead,  and  therefore  Eq.  78  to 
80,  although  having  the  advantage  of  extreme  simplicity,  can- 
not be  used  for  point-switches  without  material  error.  The 
effect  of  the  straight  frog-rail  is  less,  and  since  it  can  be  mate- 
rially reduced  by  bending  the  free  end  of  the  frog- rails,  the  in- 
fluence of  this  feature  is  frequently  ignored,  the  frog-rails  are 
assumed  to  be  curved  and  Eq.  85  and  86  are  used.  (See  §  276 
for  a  further  discussion  of  this  point.) 


284 


RAILROAD   CONSTRUCTION. 


§267. 


267.  Dimensions  for  a  turnout  from  the  OUTER  side  of  a  curved 
track.  In  this  demonstration  the  switch-rails  will  be  considered 
as  uniformly  circular  frfrm  the  switch-points  to  the  frog-point. 


FIG.  141. 
In  the  triangle  FCD  (Fig.  141)  we  have 

(FC+  CD) :  (FC-  CD) : :  tan  $(FDC+DFC) :  tan  i(FDC-DFC) ; 
but  %(FDC+  DFC)  =  90°  -  $6 

and  \(FDC  -  DFC)  =  ±F. 

Also          FG -I-  CD  =  272     and     FC  -  CD  =  a: 

7 :  :  cot  4-0 :  tan 
: :  cot 


~. 


Also 


FC\  :  sin  6>  :  sin  0  ; 


=  Z  = 


but 


=  (^-  *); 


(91) 


(92) 


sn 


If  the  curvature  of  the  main  track  is  very  sharp  or  the  frog 
angle  unusually  small,  F  may  be  less  than  0;  in  which  case  the 
center  0  will  be  on  the  same  side  of  the  main  track  as  C.  Eq. 
92  will  become  (by  calling  r  =  —  r  and  changing  the  signs) 


§267. 


SWITCHES  AND  CROSSINGS. 


285 


If  we  call  d  the  degree  of  curve  corresponding  to  the  radius 
r,  and  D  the  degree  of  curve  corresponding  to  the  radius  R,  also 
d '  the  degree  of  curve  of  a  turnout  from  a  straight  track  (the 
frog  angle  F  being  the  same),  it  may  be  shown  that  d  =  d '  — -  D 
(very  nearly).  To  illustrate  we  will  take  three  cases,  a  number 
6  frog  (very  blunt),  a  number  9  frog  (very  commonly  used),  and 
a  number  12  frog  (unusually  sharp).  Suppose  D  =  4°  0';  also 
D  =  10°  0';  g  =  4'  8J"  =  4/.708. 


Frog 
number. 

Z>  =  4°. 

"  L  "  for 
straight  track. 

d 

d'  —  D 

Error. 

L 

6 
9 
12 

12°  54'   20" 
3    30    27 
0    13    33 

12°  57'   52" 
3    31    04 
0    13    36 

0°  03'    32" 
0      0    37 
0      0    03 

56.57 

84.85 
112.72 

56.50 

84.75 
113.00 

Frog 
number. 

Z>=  10° 

"£"for 
straight  track. 

d 

d'  —  D 

Error. 

L 

6 
9 

12 

6°  53'  24" 
2    27    54 
5    44    26 

6°  57'    52" 
2    28    56 
5    46    24 

0°  04'   28" 
0    01     02 
0    01     58 

56.66 
84.86 
112.91 

56.50 
84.75 
113.00 

A  brief  study  of  the  above  tabular  form  will  show  that  the 
error  involved  in  the  use  of  the  approximate  rule  for  ordinary 
curves  (4°  or  less)  and  for  the  usual  frogs  (about  R"o.  9)  is  really 
insignificant,  and  that,  even  for  sharper  curves  (10°  or  more), 
or  for  very  blunt  frogs,  the  error  would  never  cause  damage, 
considering  the  lower  probable  speed.  In  the  most  unfavorable 
case  noted  above  the  change  in  radius  is  about  1$.  On  account 
of  the  closeness  of  the  approximation  the  method  is  frequently 
used.  The  remarkable  agreement  of  the  computed  values  of  L 
with  the  corresponding  values  for  a  straight  main  track  (the  lead 


286 


BAILROAZ)   CONSTRUCTION. 


§268. 


rails  circular  throughout)  shows  that  the  error  is  insignificant  in 
using  the  more  easily  computed  values. 

268.  Dimensions  for  a  turnout  from  the  INNER  side  of  a  curved 


track,     (Lead  rails  circular  throughout.)     From  Fig.  142    we 
have 


but 
and 


FDC)  =  90°  - 


-  FDC)  = 
\g\  :cot    Q\  tan 
:  :cot 


(95) 


OF:  FC\  :  sin  6  :  sin  (F  +  0). 

+  to  =  (B  - 


=  BF  =Z(E  -  \g)  sin 


.     ,     (96) 
-    -     (97) 


As  in  §  267,  it  may  be  readily  shown  that  the  degree  of  the 
turnout  (d)  is  nearly  the  sum  of  the  degree  of  the  main  track 
(D)  and  the  degree  (d ')  of  a  turnout  from  a  straight  track  when 
the  frog  angle  is  the  same.  The  discrepancy  in  this  case  is 


SWITCHES  AND  CROSSINGS. 


287 


somewhat  greater  than  in  the  other,  especially  when  the  curva- 
ture of  the  main  track  is  sharp.  If  the  frog  angle  is  also  large, 
the  curvature  of  the  turnout  is  excessively  sharp.  If  the  frog 
angle  is  very  small,  the  liability  to  derailment  is  great.  Turn- 
outs to  the  inside  of  a  curved  track  should  therefore  be  avoided, 
unless  the  curvature  of  the  main  track  is  small. 

269.  Double  turnout  from  a  straight  track.     In  Fig.  143  the 
frogs  FI  and  Fr  are  generally  made  equal.     Then,  if  there  are 


FIG.  143. 


uniform  curves  from  B'  to  Fl  and  from  B  to  Fr,  the  required 
value  of  Fm  is  obtained  from 


'   '  '  •  •  (98) 


r  being  found  from  Eq.  78,  in  which  n  is  the  frog  number 
of  Fl  or  Fr. 

MFm  =  rta 


but  since  nm  =  £  cot 


(99) 


Since  vers  F,  = 


vers       m  =      vers 


(100) 


288  BAILBOAD  COJfSTBUGTIOJf.  §  269. 

Also,  since  (<W  =  (MFm}'  +  (G,M)',  we  have 


r'  +  rg  +  \<f  =  ^  +  r\ 
Simplifying  and  substituting  r  =  Sgn*,  we  have 


n 


nm    = 


Dropping  the  J,   which  is  always  insignificant  in  comparison 
with  2tt2,  we  have 

-  =  n  X  .707  (approx.).       .     .   (101) 


Frogs  are  usually  made  with  angles  corresponding  to  integral 
values  of  n,  or  sometimes  in  u  half  "  sizes,  e.g.  6,  6J-,  7,  7-J,  etc. 
If  No.  8J  frogs  are  used  for  FI  and  Fr  ,  the  exact  frog  number 
for  Fm  is  6.01.  This  is  so  nearly  6  that  a  No.  6  frog  may  be 
used  without  sensible  inaccuracy.  Numbers  7  and  10  are  a 
less  perfect  combination.  If  sharp  frogs  must  be  used,  8J  and 
12  form  a  very  good  combination. 

If  it  becomes  necessary  to  use  other  frogs  because  the  right 
combination  is  unobtainable,  it  may  be  done  by  compounding 
the  curve  at  the  middle  frog.  Ft  and  Fr  should  be  greater 
than  \Fm.  If  equal  to  \Fm  ,  the  rails  would  be  straight  from 
the  middle  frog  to  the  outer  frogs.  In  Fig.  144,  S^=.Fl~ 
Drawing  the  chord 


§270. 


SWITCHES  AND  CROSSINGS. 
KF~ 


289 


sn 


KF,  =  KFm  cot 


2 •  ClOSI 

•in  K-Fi +iF.)"  (      ' 


;    (103) 


=       cot 


4  x  > 

~-  2  sin  *0  ~  4 


sn 


sn 


(104) 


FIG.  144. 

If  three  frogs,  all  different,  must  be  used,  the  largest  may  be 
selected  as  Fm ;  the  radius  of  the  lead  rails  may  be  found  by  an 
inversion  of  Eq.  98;  Fm  may  be  located  in  the  center  of  the 
tracks  by  Eq.  99 ;  then  each  of  the  smaller  frogs  may  be  located 
by  separate  applications  of  Eq.  102  or  103,  the  radius  being 
determined  by  Eq.  104. 

270.  Two  turnouts  on  the  same  side.  In  Fig.  145,  let 
0,  bisect  6>,Z>.  Then  (r,  +  ±g)  =  ifc  •+  &)  5  also,  0,0,  =  0^ 
and  Fr  =  Ft. 


vers  ^rm  = 


(105) 


(106) 


It  may  readily  be  shown  that  the  relative  values  of 


and  Fm  are  almost  identical  with  those  given  in  §  269  ;  as  may 


290 


BAILROAD  CONSTRUCTION. 


§271. 


be  apparent  when  it  is  considered  that  the  middle  switch  may 
be   regarded   simply   as   a   curved   main   track,    and   that,   as 


developed  in  §  267,  the  dimensions  of  turnouts  are  nearly  the 
same  whether  the  main  track  is  straight  or  slightly  curved. 

271.  Connecting  curve   from    a  straight  track.     The  ''con- 

necting curve  '  '  is  the  track  lying 
between  the  frog  and  the  side 
track  where  it  becomes  parallel 
to  the  main  track  (FS  in  Fig. 
146  or  147).  Call  d  the  distance 
between  track  centers.  The  angle 
FO,R  =  F  (see  Fig.  146).  Call 
T'  the  radius  of  the  connecting 
curve.  Then 


FIG.  146. 


(107) 


FR  =  (rr  --  $g)  sin  F.       .      .   (108) 

If  it  is  considered  that  the  distance  FR  consumes  too  much 
track  room,  it  may  be  shortened  by  the  method  indicated  in 
Fig.  151. 

272.  Connecting  curve  from  a  curved  track  to  the  OUTSIDE. 
When  the  main  track  is  curved,  the  required  quantities  are  the 
radius  T  of  the  connecting  curve  from  F  to  S,  Fig.  147,  and  its 
length  or  central  angle.  In  the  triangle  CSF 

CS+CF:  CS-CF  -.:  t 


§273. 


SWITCHES  AND  CROSSINGS. 


291 


but  ±(CFS  +  CSF)  =  90  -  ^>;  and,  since  the  triangle  O. 
is  isosceles,  %(CFS  -  CSF)  = 


d  :  d  —  g  ::  cot  i#  :  tan 


FIG.  147. 
From  the  triangle  CO,F  we  may  derive 

r  -  \g  :  E  +  \g  ::  sin  ^  :  sin 


Also 


(109) 


.      .     .     .     (Ill) 


273,  Connecting  curve  from  a  curved  track  to  the  INSIDE. 

As  above,  it  may  readily  be  deduced  from  the  triangle  CFS  (see 
Fig.  148)  that 

d:d  —       i:  cot        :  tan 


and  finally  that 


292  RAILROAD  CONSTRUCTION. 

Similarly  we  may  derive  (as  in  Eq.  110) 


Also 


=     r  -        sn          - 


§273. 

.     .     (113) 
.     .     (114) 


...T.  FIG.  148. 

Two  other  cases  are  possible,     (a)  r  may  increase  until  it 

becomes  infinite  (see  Fig.  149),  then 
F  =  ip.  In  such  a  case  we  may 
write,  by  substituting  in  Eq.  112, 

%R  —  d  =  4:n*(d  —  g).  .     .     (115) 

This  equation  shows  the  value  of 
R,  which  renders  this  case  possible 
with  the  given  values  of  n,  d,  and 
g.  (b)  $  may  be  greater  than  F. 
As  before  (see  Fig.  150) 


—  d  :  d  -  g  :  :  cot 


:  tan 


FIG.  149. 


-  g) 


§  274.  SWITCHES  AND  CROSSINGS. 

the  same  as  Eq.  112,  but 

sin  0 


293 


FIG.  150. 

274.  Crossover  between  two  parallel  straight  tracks.  (See 
Fig.  151.)  The  turnouts  are  as  usual.  The  crossover  track  may 
be  straight,  as  shown  by  the  full  02 
lines,  or  it  may  be  a  reversed 
curve,  as  shown  by  the  dotted 
lines.  The  reversed  curve  short- 
ens the  total  length  of  track  re- 
quired, but  is  somewhat  objection- 
able. The  first  method  requires 
that  both  frogs  must  be  equal. 
The  second  method  permits  un- 
equal frogs,  although  equal  frogs 
are  preferable.  The  length  of 
straight  crossover  track  is 


FIG.  151. 


Fl  +  g  cos  F,  =  d  -  g\ 


sin 


(117) 


294  RAILROAD   CONSTRUCTION.  §  274. 

The  total  distance  along  the  track  may  be  derived  as  follows  : 

D  V  =  ZDF,  +  F,Y=  %DF,  +  XT-  XF,  ; 

XY=  (d  -  g)  cot  F,  ;         XF*  =  g  -j-  sin  F^  ; 

(d-g)cotFl--.    .     .     (118) 


If  a  reversed  curve  with  equal  frogs  is  used,  we  have 


also 


2r 
DQ  =  2r  siii  0. 


o2 
FIG.  152. 

If  the  frogs  are  unequal,  we  will  have  (see  Fig.  152) 
r,  vers  6  +  r1  vers  0  =  d ; 

vers  0  =  — q^ —  ;    . 

also  the  distance  along  the  track 

JB*N  =  (r.  -4-  7*2)  sin  6.     .     .     . 


(119) 
(120) 


(121) 


(122) 


§275. 


SWITCHES  AND  CROSSINGS. 


295 


275.  Crossover  between  two  parallel  curved  tracks,  (a)  Using 
a  straight  connecting  curve.  This  solution  has  limitations.  If 
one  frog  (Ft)  is  chosen,  F^  becomes  determined,  being  a  function 
of  Ft.  If  FI  is  less  than  some  limit,  depending  on  the  width 


FIG.  153. 


(d)  between  the  parallel  tracks,  this  solution  becomes  impossible. 
In  Fig.  153  assume  F,  as  known.  Then  F,H  '=  g  sec  F,.  In 
the  triangle  HOF^  we  have 


sin  HF,0  :  sin  F,HO  ::  HO  : 
sin  F,HO  =  cos  F,  ;     HF,0  =  90°  + 
.-.    sin  HF^O  =  cos  Fv 


-       -     sec 


(123) 


296 


RAILROAD   CONSTRUCTION. 


§275. 


Knowing  F^  0,  is  determinable  from  Eq.  91.  Fig.  153  shows 
the  case  where  0a  is  greater  than  F^.  Fig.  154  shows  the  case 
where  it  is  less.  The  demonstration  of  Eq,  123  is  applicable  to 


FIG.  154. 

both  figures.  The  relative  position  of  the  frogs  F,  and  Ft  may 
be  determined  as  follows,  the  solution  being  applicable  to  both 
Figs.  153  and  154: 

HOF,  =  180°  -  (90°  -  F,)  -  (90°  +  F,)  =  F,  -  F,. 
Then 

>;).     .     .     ,     (124) 


Since  Ft  comes  out  any  angle,  its  value  will  not  be  in  general 
that  of  an  even  frog  number,  and  it  will  therefore  need  to  be 
made  to  order. 

(b)  Continuing  the  switch-rail  curves  until  they  meet  as  a 
reversed  curve.  In  this  case  F^  and  F^  may  be  chosen  at  pleasure 
(within  limitations),  and  they  will  of  course  be  of  regular  sizes 
and  equal  or  unequal  as  desired.  F1  and  F9  being  known,  0, 
and  0,  are  computed  by  Eq.  95  and  91.  In  the  triangle 
(see  Fig.  155) 


vers  ib  = 


_  00.)  (ff-  00,) 
00,  -  00, 


in  which 


S=  ^(00, +  00, +  0,0,); 


§  275. 

but 


SWITCHES  AND  CROSSINGS. 
t  =  Jt  +  id  —  r^ 


297 


_  00t  =  B  +  r,  -  R  - 


FIG.  155. 


vers      = 


1  = 


3  =  2(72  -  JrZ  +  fer)  sin  £(•/,  -  ^  _ 


(125) 

(126) 

(127) 
(128) 


RAILROAD  CONSTRUCTION. 


§276. 


Although  the  above  method  introduces  a  reversed  curve,  yet 
it  uses  up  less  track  than  the  first  method  and  permits  the  use  of 
ordinary  frogs  rather  than  those  having  some  special  angle  which 
must  be  made  to  order. 

276.  Practical  rules  for  switch-laying.  A  consideration  of 
the  previous  sections  will  show  that  the  formulae  are  compara- 
tively simple  when  the  lead  rails  are  assumed  as  circular ;  that 
they  become  complicated,  even  for  turnouts  from  a  straight 
main  track,  when  the  effect  of  straight  frog  and  point  rails  is 
allowed  for,  and  that  they  become  hopelessly  complicated  when 
allowing  for  this  effect  on  turnouts  from  a  curved  main  track. 
It  is  also  shown  (§  267)  that  the  length  of  the  lead  is  practically 


FIG.  140. 

the  same  whether  the  main  track  is  straight  or  is  curved  with 
such  curves  as  are  commonly  used,  and  that  the  degree  of  curve 
of  the  lead  rails  from  a  curved  main  track  may  be  found  with 
close  approximation  by  mere  addition  or  subtraction.  From 
this  it  may  be  assumed  that,  if  the  length  of  lead  (L]  and  the 
radius  of  the  lead  rails  (r)  are  computed  from  Eq.  87  and  90  for 
various  frog  angles,  the  same  leads  may  be  used  for  curved  main 
track ;  also,  that  the  degree  of  curve  of  the  lead  rails  may  be 
found  by  addition  or  subtraction,  as  indicated  in  §  267,  and  that 
the  approximations  involved  will  not  be  of  practical  detriment. 


§  276.  SWITCHES  AND  CROSSINGS.  299 

In  accordance  with  this  plan  Table  III  has  been  computed  from 
Eq.  87,  88,  and  90  A  The  leads  there  given  may  be  used  for  all 
main  tracks  straight  or  curved.  The  table  gives  the  degree  of 
curve  of  the  lead  rails  for  straight  main  track ;  for  a  turnout  to 
the  inside,  add  the  degree  of  curve  of  the  main  track ;  for  a 
turnout  to  the  outside,  subtract  it.  ft 

If  the  position  of  the  switch  -block  is  definitely  determined, 
then  the  rails  must  be  cut  accordingly ;  but  when  some  freedom 
is  allowable  (which  never  need  exceed  15  feet  and  may  require 
but  a  few  inches),  one  rail-cutting  may  be  avoided.  Mark  on 
the  rails  at  B,  F,  and  D ;  measure  off  the  length  of  the  switch- 
rails  DN\  offset  \g-\-~k  from  N  for  the  point  S. 
The  point  H  may  be  located  (temporarily)  by  meas- 
uring along  the  rail  a  distance  FH  (=  f)  and  then 
swinging  out  a  distance  of  f  -r-  n  (n  being  the  frog 
number).  JIT  =  \g  and  is  measured  at  right 
angles  to  FH.  Points  for  track  centers  between  S 
and  T  may  be  laid  off  by  a  transit  or  by  the  use  of  a 
string  and  tape.  Substituting  in  Eq.  31  the  value 
of  R  and  of  chord  (=  ST),  we  may  compute  x  (= 
db).  Locate  the  middle  point  d  and  the  quarter 
points  a"  and  c" '.  Then  a" a  and  c"c  each  equal  FlG- 156' 
three-fourths  of  db.  Theoretically  this  gives  a  parabola  rather 
than  a  circle,  but  the  difference  for  all  practical  cases  is  too 
small  for  measurement. 

Example.  Given  a  main  track  on  a  4°  curve ;  a  turnout  to 
the  outside,  using  a  number  9  frog;  gauge  4'  8J";  f  =  3'.  37; 
k  =  5f" ;  DN  =  15'  0"  and  a  =  1°  50'.  Then  for  a  straight 
track  r  would  equal  681.16  \d' =  8°  25'].  For  this  curved 
track  d  will  be  nearly  (8°  25'  -  4°)  =  4°  25',  or  r  will  be 
1297.6.  L  for  the  straight  track  would  be  72.20;  but  since 
the  lead  is  slightly  increased  (see  §  267)  when  the  turnout  is  on 
the  outside  of  a  curve,  L  may  here  be  called  72.5.  FH  =  f 
=  3'.37  ;/-*-»  =  3.37  -=-  9  =  0'.375  =  4".5.  H,  T,  and  S 
may  be  located  as  described  above.  ST  may  be  measured  on  the 
ground,  or  it  may  be  computed  from  Eq.  88,  giving  the  value 


300 


RAILROAD   CONSTRUCTION. 


277. 


of  53.80  feet  for  straight  track.  Since  it  is  slightly  more  for  a 
turnout  to  the  outside  of  a  curve,  it  may  be  called  54.0.  Then 
(54. 0)3 


x  =  db  = 
foot. 


8  X   1297.6 


=  0.281  feet,  and  aa"  and  cc"  =  0.21 


CROSSINGS. 


277.  Two  straight  tracks.  When  two  straight  tracks  cross 
each  other,  four  frogs  are  necessary,  the  angles  of  two  of  them 
being  supplementary  to  the  angles  of  the  other.  Since  such 
crossings  are  sometimes  operated  at  high  speeds,  they  should  be 


SECTION  ON  A-B 


SECTIONVON  C-D 
FIG.  157.— CROSSING. 


very  strongly  constructed,  and  the  angles  should  preferably  be 
90°  or  as  near  that  as  possible.  The  frogs  will  not  in  general 
be  "  stock"  frogs  of  an  even  number,  especially  if  the  angles 
are  large,  but  must  be  made  to  order  with  the  required  angles 
as  measured.  In  Fig.  157  are  shown  the  details  of  such  a  cross- 
ing. Note  the  fillers,  bolts,  and  guard-rails. 


UNIVERSITY 


279. 


SWITCHES  AND  CROSSINGS. 


301 


278.  One  straight  and  one  curved  track.     Structurally  the 

crossing  is  about  the  same  as  above,  but 

the  frog  angles  are 

all  unequal.     In  Fig.  158,  R  is  known, 

and  the   angle  J/,  made  by  the  center 

j 

lines  of  the  tracks  at  their  point  of  inter- 

P 

section,  is  also  known. 

-4 

•<L 

p 

_ 

..^O 

Mj^ 

rs 

M  =  NCM.     NC  =  R  cos  M. 

/ 

^CJM  / 

X. 

"—  X 

p    ,'\  i^.-. 

N. 

(R—%g)  cos  FI  =  NC  +  4#  ; 

/     / 

/^ 

^vO\ 

i/~ 

/     / 

1  '   '       1 

F!    \      N 

p,  -ti  cos  JxL-\-\g 

•  (129) 

1 

//// 

'V 

/  \ 

V 

C°S  ^B^Tx+to 

Similarly  cos  I?  ,  —        T?_J_JL        > 

R  cos  M—  \q 
cos  F  — 

//,'/    / 

V 

< 

t  
c 

—  -JN 

R  cos  M—  \g 

/»/-ko      H    —   —  — 

• 

R-\g 


FIG.  158. 


279.  Two  curved  tracks.     The  four  frogs  are  unequal,  and 
the  angle  of  each  must  be  computed.     The  radii  Rl  and  ^a  are 


FIG.  159, 


known ;   also  the   angle  M.     r^,  r3 ,   r, ,   and  r4  are  therefore 
known  by  adding  or  subtracting  ^,  but  the  lines  are  so  indi- 


302 


RAILROAD  CONSTRUCTION. 


279. 


cated  for  brevity.     Call  the   angle  MC,C,  =  Ciy   the    angle 
MCtCt  =  <72,  and  the  line  C&  =  c.     Then 


and 


tan  J(0;  -  tf.)  =  cot 


(7,  and  (73  then  become  known  and 


sn 


In  the  triangle  F,  C,  Ct ,  call  %(c  +  r,  +  r4)  =  Sl ;  then 
vers  Fv  =       l  ~  r'^Sl  ~~  r*) 


Similarly  vers  F  = 


vers  ^.  = 


(130) 


In  the  above  equations 


APPENDIX. 


THE  ADJUSTMENTS  OF  INSTEUMENTS. 


THE  accuracy  of  instrumental  work  may  be  vitiated  by  any 
one  of  a  large  number  of  inaccuracies  in  the  geometrical  rela- 
tions of  the  parts  of  the  instruments.  Some  of  these  relations 
are  so  apt  to  be  altered  by  ordinary  usage  of  the  instrument  that 
the  makers  have  provided  adjusting-screws  so  that  the  inaccura- 
cies may  be  readily  corrected.  There  are  other  possible  defects, 
which,  however,  will  seldom  be  found  to  exist,  provided  the 
instrument  was  properly  made  and  has  never  been  subjected  to 
treatment  sufficiently  rough  to  distort  it.  Such  defects,  when 
found,  can  only  be  corrected  by  a  competent  instrument  maker 
or  repairer. 

A  WARNING  is  necessary  to  those  who  would  test  the  accuracy 
of  instruments,  and  especially  to  those  whose  experience  in  such 
work  is  small.  Lack  of  skill  in  handling  an  instrument  will 
often  indicate  an  apparent  error  of  adjustment  when  the  real 
error  is  very  different  or  perhaps  non-existent.  It  is  always  a 
safe  plan  when  testing  an  adjustment  to  note  the  amount  of  the 
apparent  error ;  then,  beginning  anew,  make  another  independ- 
ent determination  of  the  amount  of  the  error.  When  two  or 
more  perfectly  independent  determinations  of  such  an  error  are 
made  it  will  generally  be  found  that  they  differ  by  an  appreciable 
amount.  The  differences  may  be  due  in  variable  measure  to 
careless  inaccurate  manipulation  and  to  instrumental  defects 
which  are  wholly  independent  of  the  particular  test  being  made. 
Such  careful  determinations  of  the  amounts  of  the  errors  are 
generally  advisable  in  view  of  the  next  paragraph. 

303 


304  THE  ADJUSTMENTS  OF  INSTRUMENTS. 

Do     NOT    DISTURB     THE     ADJUSTING -SCREWS     ANY    MORE     THAN 

NECESSARY.  Although  metals  are  apparently  rigid,  they  are 
really  elastic  and  yielding.  If  some  parts  of  a  complicated 
mechanism,  which  is  held  together  largely  by  friction,  are  sub- 
jected to  greater  internal  stresses  than  other  parts  of  the  mech- 
anism, the  jarring  resulting  from  handling  will  frequently  cause 
a  slight  readjustment  in  the  parts  which  will  tend  to  more  nearly 
equalize  the  internal  stresses.  Such  action  frequently  occurs 
with  the  adjusting  mechanism  of  instruments.  One  screw  may 
be  strained  more  than  others.  The  friction  of  parts  may  pre- 
vent the  opposing  screw  from  immediately  taking  up  an  equal 
stress.  Perhaps  the  adjustment  appears  perfect  under  these 
conditions.  Jarring  diminishes  the  friction  between  the  parts, 
and  the  unequal  stresses  tend  to  equalize.  A  motion  takes  place 
which,  although  microscopically  minute,  is  sufficient  to  indicate 
an  error  of  adjustment.  A  readjustment,  made  by  unskillful 
hands,  may  not  make  the  final  adjustment  any  more  perfect. 
The  frequent  shifting  of  adjusting-screws  wears  them  badly, 
and  when  the  screws  are  worn  it  is  still  more  difficult  to  keep 
them  from  moving  enough  to  vitiate  the  adjustments.  It  is 
therefore  preferable  in  many  cases  to  refrain  from  disturbing  the 
adjusting-screws,  especially  as  the  accuracy  of  the  work  done  is 
not  necessarily  affected  by  errors  of  adjustment,  as  may  be  illus- 
trated : 

(a)  Certain  operations  are  absolutely  unaffected  by  certain 
errors  of  adjustment. 

(b)  Certain  operations  are  so  slightly  affected  by  certain  small 
errors  of  adjustment  that  their  effect  may  properly  be  neglected. 

(G)  Certain  errors  of  adjustment  may  be  readily  allowed  for 
and  neutralized  so  that  no  error  results  from  the  use  of  the  un- 
adjusted instrument.  Illustrations  of  all  these  cases  will  be 
given  under  their  proper  heads. 

ADJUSTMENTS    OF    THE    TRANSIT. 

1.  To  have  the  plate-bubbles  in  the  center  of  the  tubes  when 
the  axis  is  vertical.  Clamp  the  upper  plate  and,  with  the  lower 


THE  ADJUSTMENTS  OF  INSTRUMENTS.  305 

<3lamp  loose,  swing  the  instrument  so  that  the  plate-bubbles  are 
parallel  to  the  lines  of  opposite  leveling-screws.  Level  up  until 
both  bubbles  are  central.  Swing  the  instrument  180°.  If  the 
bubbles  again  settle  at  the  center,  the  adjustment  is  perfect.  If 
either  bubble  does  not  settle  in  the  center,  move  the  leveling- 
screws  until  the  bubble  is  half-way  back  to  the  center.  Then, 
before  touching  the  adjusting-screws,  note  carefully  the  position 
of  the  bubbles  and  observe  whether  the  bubbles  always  settle  at 
the  same  place  in  the  tube,  no  matter  to  what  position  the  in- 
strument may  be  rotated.  When  the  instrument  is  so  leveled, 
the  axis  is  truly  vertical  and  the  discrepancies  between  this  con- 
stant position  of  the  bubbles  and  the  centers  of  the  tubes  measure 
the  errors  of  adjustment.  By  means  of  the  adjusting-screws 
bring  each  bubble  to  the  center  of  the  tube.  If  this  is  done  so 
skillfully  that  the  true  level  of  the  instrument  is  not  disturbed, 
the  bubbles  should  settle  in  the  center  for  all  positions  of  the 
instrument.  Under  unskillful  hands,  two  or  more  such  trials 
may  be  necessary. 

When  the  plates  are  not  horizontal,  the  measured  angle  is  greater  than 
the  true  horizontal  angle  by  the  difference  between  the  measured  angle 
and  its  projection  on  a  horizontal  plane.  When  this  angle  of  inclination 
is  small,  the  difference  is  insignificant.  Therefore  when  the  plate-bubbles 
are  very  nearly  in  adjustment,  the  error  of  measurement  of  horizontal 
angles  may  be  far  within  the  lowest  unit  of  measurement  used.  A  small 
error  of  adjustment  of  the  plate-bubble  perpendicular  to  the  telescope  will 
affect  the  horizontal  angles  by  only  a  small  proportion  of  the  error,  which 
will  be  perhaps  imperceptible.  Vertical  angles  will  be  affected  by  the 
same  insignificant  amount.  A  small  error  of  adjustment  of  the  plate- 
bubble  parallel  to  the  telescope  will  affect  horizontal  angles  very  slightly, 
but  will  affect  vertical  angles  by  the  full  amount  of  the  error. 

All  error  due  to  unadjusted  plate-bubbles  may  be  avoided  by  noting  in 
what  positions  in  the  tubes  the  bubbles  will  remain  fixed  for  all  positions 
of  azimuth  and  then  keeping  the  bubbles  adjusted  to  these  positions,  for 
the  axis  is  then  truly  vertical.  It  will  often  save  time  to  work  in  this  way 
temporarily  rather  than  to  stop  to  make  the  adjustments.  This  should 
especially  be  done  when  accurate  vertical  angles  are  required. 

When  the  bubbles  are  truly  adjusted,  they  should  remain  stationary, 
regardless  of  whether  the  telescope  is  revolved  with  the  upper  plate  loose 
and  the  lower  plate  clamped  or  whether  the  whole  instrument  is  revolved, 
the  plates  being  clamped  together.  If  there  is  any  appreciable  difference, 


306  THE  ADJUSTMENTS  OF  INSTRUMENTS. 

it  shows  that  the  two  vertical  axes  or  "  centers  "  of  the  plates  are  not  con- 
centric. This  may  be  due  to  cheap  and  faulty  construction  or  to  the  exces- 
sive wear  that  may  be  sometimes  observed  in  an  old  instrument  originally 
well  made.  In  either  case  it  can  only  be  corrected  by  a  maker. 

2.  To  make  the  revolving  axis  of  the  telescope  perpendicular 
to  the  vertical  axis  of  the  instrument.     This  is  best  tested  by 
using  a  long  plumb-line,  so  placed  that  the  telescope  must  be 
pointed  upward  at  an  angle  of  about  45°  to  sight  at  the  top  of 
the  plumb-line  and  downward  about  the  same  amount,  if  pos- 
sible, to  sight  at  the  lower  end.     The  vertical  axis  of  the  transit 
must  be  made  truly  vertical.     Sight  at  the  upper  part  of  the 
line,  clamping  the  horizontal  plates.     Swing  the  telescope  down 
and  see  if  the  cross-wire  again  bisects  the  cord.      If  so,  the 
adjustment  is  probably  perfect  (a  conceivable  exception  will  be 
noted  later) ;  if  not,  raise  or  lower  one  end  of  the  axis  by  means 
of  the  adjusting-screws,  placed  at  the  top  of  one  of  the  stan- 
dards, until  the  cross-wire  will  bisect  the  cord  both  at  top  and 
bottom.      The  plumb-bob  may  be  steadied,  if   necessary,   by 
hanging  it  in  a  pail  of  water.     As  many  telescopes  cannot  be 
focused  on  an  object  nearer  than  6  or  8  feet  from  the  telescope, 
this  method  requires  a  long  plumb-line  swung  from  a  high  point, 
which  may  be  inconvenient. 

Another  method  is  to  set  up  the  instrument  about  10  feet 
from  a  high  wall.  After  leveling,  sight  at  some  convenient 
mark  high  up  on  the  wall.  Swing  the  telescope  down  and  make 
a  mark  (when  working  alone  some  convenient  natural  mark  may 
generally  be  found)  low  down  on  the  wall.  Plunge  the  telescope 
and  revolve  the  instrument  about  its  vertical  axis  and  again  sight 
at  the  upper  mark.  Swing  down  to  the  lower  mark.  If  the 
wire  again  bisects  it,  the  adjustment  is  perfect.  If  not,  fix  a 
point  half-way  between  the  two  positions  of  the  lower  mark. 
The  plane  of  this  point,  the  upper  point,  and  the  center  of  the 
instrument  is  truly  vertical.  Adjust  the  axis  to  these  upper  and 
lower  points  as  when  using  the  plumb-line. 

3.  To  make  the  line  of  collimation  perpendicular  to  the 
revolving  axis  of  the  telescope*     With  the  instrument  level  and 


THE  ADJUSTMENTS  OF  INSTRUMENTS.  307 

the  telescope  nearly  horizontal  point  at  some  well-defined  point 
at  a  distance  of  200  feet  or  more.  Plunge  the  telescope  and 
establish  a  point  in  the  opposite  direction.  Turn  the  whole 
instrument  about  the  vertical  axis  until  it  again  points  at  the 
first  mark.  Again  plunge  to  "  direct  position"  (i.e.,  with  the 
level-tube  under  the  telescope).  If  the  vertical  cross-wire  again 
points  at  the  second  mark,  the  adjustment  is  perfect.  If  not, 
the  error  is  one-fourth  of  the  distance  between  the  two  positions 
of  the  second  mark.  Loosen  the  capstan-screw  on  one  side  of 
the  telescope  and  tighten  it  on  the  other  side  until  the  vertical 
wire  is  set  at  the  one-fourth  mark.  Turn  the  whole  instrument 
by  means  of  the  tangent  screw  until  the  vertical  wire  is  midway 
between  the  two  positions  of  the  second  mark.  Plunge  the 
telescope.  If  the  adjusting  has  been  skillfully  done,  the  cross- 
wire  should  come  exactly  to  the  first  mark.  As  an  "erecting 
eyepiece  "  rein  verts  an  image  already  inverted,  the  ring  carrying 
the  cross-wires  must  be  moved  in  the  same  direction  as  the 
apparent  error  in  order  to  correct  that  error. 

The  necessity  for  the  third  adjustment  lies  principally  in  the  practice 
of  producing  a  line  by  plunging  the  telescope,  but  when  this  is  required  to 
be  done  with  great  accuracy  it  is  always  better  to  obtain  the  forward  point 
by  reversion  (as  described  above  for  making  the  test)  and  take  the  mean 
of  the  two  forward  points.  Horizontal  and  vertical  angles  are  practically 
unaffected  by  small  errors  of  this  adjustment,  unless,  in  the  case  of 
horizontal  angles,  the  vertical  angles  to  the  points  observed  are  very 
different. 

Unnecessary  motion  of  the  adjusting-screws  may  sometimes  be  avoided 
by  carefully  establishing  the  forward  point  on  line  by  repeated  reversions 
of  the  instrument,  and  thus  determining  by  repeated  trials  the  exact 
amount  of  the  error.  Differences  in  the  amount  of  error  determined 
would  be  evidence  of  inaccuracy  in  manipulating  the  instrument,  and 
would  show  that  an  adjustment  based  on  the  first  trial  would  probably 
prove  unsatisfactory. 

The  2d  and  3d  adjustments  are  mutually  dependent.  If  either  adjust- 
ment is  badly  out,  the  other  adjustment  cannot  be  made  except  as 
follows  : 

(a)  The  second  adjustment  can  be  made  regardless  of  the  third  when 
the  lines  to  the  high  point  and  the  low  point  make  equal  angles  with  the 
horizontal. 


308  TEE  ADJUSTMENTS  OF  INSTRUMENTS. 

(6)  The  third  adjustment  can  be  made  regardless  of  the  second  when 
the  front  and  rear  points  are  on  a  level  with  the  instrument. 

When  both  of  these  requirements  are  nearly  fulfilled,  and  especially 
when  the  error  of  either  adjustment  is  small,  no  trouble  will  be  found  in 
perfecting  either  adjustment  on  account  of  a  small  error  in  the  other 
adjustment. 

If  the  test  for  the  second  adjustment  is  made  by  means  of  the  plumb- 
line  and  the  vertical  cross-wire  intersects  the  line  at  all  points  as  the  tele- 
scope is  raised  or  lowered,  it  not  only  demonstrates  at  once  the  accuracy 
of  that  adjustment,  but  also  shows  that  the  third  adjustment  is  either 
perfect  or  has  so  small  an  error  that  it  does  not  affect  the  second. 

4.  To  have  the  ~bubble  of  the  telescope-level  in  the  center  of 
the  tube  when  the  line  of  colliination  is  horizontal.  The  line  of 
collimation  should  coincide  with  the  optical  axis  of  the  telescope. 
If  the  object-glass  and  eyepiece  have  been  properly  centered, 
the  previous  adjustment  will  have  brought  the  vertical  cross- 
wire  to  the  center  of  the  field  of  view.  The  horizontal  cross- 
wire  should  also  be  brought  to  the  center  of  the  field  of  view, 
and  the  bubble  should  be  adjusted  to  it. 

a.  Peg  method.  Set  up  the  transit  at  one  end  of  a  nearly 
level  stretch  of  about  300  feet.  Clamp  the  telescope  with  its 
bubble  in  the  center.  Drive  a  stake  vertically  under  the  eye- 
piece of  the  transit,  and  another  about  300  feet  away.  Observe 
the  height  of  the  center  of  the  eyepiece  (the  telescope  being 
level)  above  the  stake  (calling  it  a) ;  observe  the  reading  of  the 
rod  when  held  on  the  other  stake  (calling  it  &) ;  take  the  instru- 
ment to  the  other  stake  and  set  it  up  so  that  the  eyepiece  is  ver- 
tically over  the  stake,  observing  the  height,  c ;  take  a  reading  on 
the  first  stake,  calling  it  d.  If  this  adjustment  is  perfect,  then 

a  —  d  =  b  —  <?, 

or  (a  —  d)  —  (b  —  c)  =  0. 

Call  (a  —  d)  —  (b  —  c)  =  2m. 

When  m  is  positive,  the  line  points  downward; 
"     m  "  negative,  "      u        "       upward. 


THE  ADJUSTMENTS  OF  INSTRUMENTS.  309 

To  adjust :  if  the  line  points  up,  sight  the  horizontal  cross- 
wire  (by  moving  the  vertical  tangent  screw)  at  a  point  which  is 
m  lower,  then  adjust  the  bubble  so  that  it  is  in  the  center. 

By  taking  several  independent  values  for  a,  6,  c,  and  d,  a  mean  value 
for  m  is  obtained,  which  is  more  reliable  and  which  may  save  much  un- 
necessary working  of  the  adjusting-screws. 

b.  Using  an  auxiliary  level.  When  a  carefully  adjusted 
level  is  at  hand,  this  adjustment  may  sometimes  be  more  easily 
made  by  setting  up  the  transit  and  level,  so  that  their  lines  of 
collimation  are  as  nearly  as  possible  at  the  same  height.  If  a 
point  may  be  found  which  is  half  a  mile  or  more  away  and 
which  is  on  the  horizontal  cross- wire  of  the  level,  the  horizontal 
cross- wire  of  the  transit  may  be  pointed  directly  at  it,  and  the 
bubble  adjusted  accordingly.  Any  slight  difference  in  the 
heights  of  the  lines  of  collimation  of  the  transit  and  level  (say  J") 
may  almost  be  disregarded  at  a  distance  of  ^  mile  or  more,  or, 
if  the  difference  of  level  would  have  an  appreciable  effect,  even 
this  may  be  practically  eliminated  by  making  an  estimated  allow- 
ance when  sighting  at  the  distant  point.  Or,  if  a  distant  point 
is  not  available,  a  level-rod  with  target  may  be  used  at  a  dis- 
tance of  (say)  300  feet,  making  allowance  for  the  carefully  de- 
termined difference  of  elevation  of  the  two  lines  of  collimation. 

5.  Zero  of  vertical  circle.  When  the  line  of  collimation  is 
truly  horizontal  and  the  vertical  axis  is  truly  vertical,  the  read- 
ing of  the  vertical  circle  should  be  0°.  If  the  arc  is  adjustable, 
it  should  be  brought  to  0°.  If  it  is  not  adjustable,  the  index 
error  should  be  observed,  so  that  it  may  be  applied  to  all  read- 
ings of  vertical  angles. 

ADJUSTMENTS    OF    THE    WYE    LEVEL. 

1 .  To  make  the  line  of  collimation  coincide  with  the  center 
of  the  rings.  Point  the  intersection  of  the  cross- wires  at  some 
well-defined  point  which  is  at  a  considerable  distance.  The  in- 
strument need  not  be  level,  which  allows  much  greater  liberty 
in  choosing  a  convenient  point.  The  vertical  axis  should  be 


310  THE  ADJUSTMENTS  OF  INSTRUMENTS. 

clamped,  and  the  clips  over  the  wyes  should  be  loosened  and  raised. 
Rotate  the  telescope  in  the  wyes.  The  intersection  of  the  cross- 
wires  should  be  continually  on  the  point.  If  it  is  not,  it  requires 
adjustment.  Rotate  the  telescope  180°  and  adjust  one-half  of 
the  error  by  means  of  the  capstan-headed  screws  that  move  the 
cross- wire  ring.  It  should  be  remembered  that,  with  an  erect- 
ing telescope,  on  account  of  the  inversion  of  the  image,  the  ring 
should  be  moved  in  the  direction  of  the  apparent  error.  Adjust 
the  other  half  of  the  error  with  the  leveling-screws.  Then  ro- 
tate the  telescope  90°  from  its  usual  position,  sight  accurately  at 
the  point,  and  then  rotate  180°  from  that  position  and  adjust 
any  error  as  before.  It  may  require  several  trials,  but  it  is 
necessary  to  adjust  the  ring  until  the  intersection  of  the  cross- 
wires  will  remain  on  the  point  for  any  position  of  rotation. 

If  such  a  test  is  made  on  a  very  distant  point  and  again  on  a  point  only 
10  or  15  feet  from  the  instrument,  the  adjustment  may  be  found  correct 
for  one  point  and  incorrect  for  the  other.  This  indicates  that  the  object- 
slide  is  improperly  centered.  Usually  this  defect  can  only  be  corrected  by 
an  instrument-maker.  If  the  difference  is  very  small  it  may  be  ignored, 
but  the  adjustment  should  then  be  made  on  a  point  which  is  at  about  the 
mean  distance  for  usual  practice— say  150  feet. 

If  the  whole  image  appears  to  shift  as  the  telescope  is  rotated,  it  indi- 
cates that  the  eyepiece  is  improperly  adjusted.  This  defect  is  likewise 
usually  corrected  only  by  the  maker.  It  does  not  interfere  with  instru- 
mental accuracy,  but  it  usually  causes  the  intersection  of  the  cross- wires  to 
be  eccentric  with  the  field  of  view. 

2.  To  make  the  axis  of  the  level  tube  parallel  to  the  line  of 
collimation.  Raise  the  clips  as  far  as  possible.  Swing  the  level 
so  that  it  is  parallel  to  a  pair  of  opposite  leveling-screws  and 
clamp  it.  Bring  the  bubble  to  the  middle  of  the  tube  by  means 
of  the  leveling-screws.  Take  the  telescope  out  of  the  wyes  and 
replace  it  end  for  end,  using  extreme  care  that  the  wyes  are  not 
jarred  by  the  action.  If  the  bubble  does  not  come  to  the  center, 
correct  one-half  of  the  error  by  the  vertical  adjusting-screws  at 
one  end  of  the  bubble.  Correct  the  other  half  by  the  leveling- 
screws.  Test  the  work  by  again  changing  the  telescope  end  for 
end  in  the  wyes. 


THE  ADJUSTMENTS  OF  INSTRUMENTS.  311 

Care  should  be  taken  while  making  this  adjustment  to  see 
that  the  level-tube  is  vertically  under  the  telescope.  With  the 
bubble  in  the  center  of  the  tube,  rotate  the  telescope  in  the  wyes 
for  a  considerable  angle  each  side  of  the  vertical.  If  the  first 
half  of  the  adjustment  has  been  made  and  the  bubble  moves,  it 
shows  that  the  axis  of  the  wyes  and  the  axis  of  the  level-tube 
are  not  in  the  same  vertical  plane  although  both  have  been  made 
horizontal.  By  moving  one  end  of  the  level-tube  sidewise  by 
means  of  the  horizontal  screws  at  one  end  of  the  tube,  the  two 
axes  may  be  brought  into  the  same  plane.  As  this  adjustment 
is  liable  to  disturb  the  other,  both  should  be  alternately  tested 
until  both  requirements  are  complied  with. 

By  these  methods  the  axis  of  the  bubble  is  made  parallel  to 
the  axis  of  the  wyes ;  and  as  this  has  been  made  parallel  to  the 
lines  of  collimation  by  means  of  the  previous  adjustment,  the 
axis  of  the  bubble  is-  therefore  parallel  to  the  line  of  collimation. 

3.  To  make  the  line  of  collimation  perpendicular  to  the  ver- 
tical axis.  Level  up  so  that  the  instrument  is  approximately  level 
over  both  sets  of  leveling-screws.  Then,  after  leveling  carefully 
over  one  pair  of  screws,  revolve  the  telescope  180°.  If  it  is  not 
level,  adjust  half  of  the  error  by  means  of  the  capstan-headed 
screw  under  one  of  the  wyes,  and  the  other  half  by  the  leveling- 
screws.  Reverse  again  as  a  test. 

When  the  first  two  adjustments  have  been  accurately  made,  good  level- 
ing may  always  be  done  by  bringing  the  bubble  to  the  center  by  means  of 
the  leveling-screws,  at  every  sight  if  necessary,  even  if  the  third  adjust- 
ment is  not  made.  Of  course  this  third  adjustment  should  be  made  as  a 
matter  of  convenience,  so  that  the  line  of  collimation  may  be  always  level 
no  matter  in  what  direction  it  may  be  pointed,  but  it  is  not  necessary  to 
stop  work  to  make  this  adjustment  every  time  it  is  found  to  be  defective. 

ADJUSTMENTS    OF    THE    DUMPY    LEVEL. 

1 .  To  make  the  axis  of  the  level-tube  perpendicular  to  the 
vertical  axis.  Level  up  so  that  the  instrument  is  approximately 
level  over  both  sets  of  leveling-screws.  Then,  after  leveling 
carefully  over  one  pair  of  screws,  revolve  the  telescope  180°.  If 


312  THE  ADJUSTMENTS  OF  INSTRUMENTS. 

it  is  not  level,  adjust  one-half  of  the  error  by  means  of  the 
adjusting-screws  at  one  end  of  the  bubble,  and  the  other  half 
by  means  of  the  leveling-screws.  Reverse  again  as  a  test. 

2.  To  make  the  line  of  collimation  perpendicular  to  the  ver- 
tical axis.  The  method  of  adjustment  is  identical  with  that  for 
the  transit  (No.  4,  p.  308)  except  that  the  cross-wire  must  be 
adjusted  to  agree  with. the  level-bubble  rather  than  vice  versa,  as 
is  the  case  with  the  corresponding  adjustment  of  the  transit ; 
i.e.,  with  the  level-bubble  in  the  center,  raise  or  lower  the 
horizontal  cross- wire  until  it  points  at  the  mark  known  to  be  on 
a  level  with  the  center  of  the  instrument. 

If  the  instrument  has  been  well  made  and  has  not  been  dis- 
torted by  rough  usage,  the  cross-wires  will  intersect  at  the 
center  of  the  field  of  view  when  adjusted  as  described.  If  they 
do  not,  it  indicates  an  error  which  ordinarily  can  only  be  cor- 
rected by  an  instrument-maker.  The  error  may  be  due  to  any 
one  of  several  causes,  which  are 

(a)  faulty  centering  of  object-slide ; 

(£)  faulty  centering  of  eyepiece ; 

(c)  distortion  of  instrument  so  that  the  geometric  axis  of 
the  telescope  is  not  perpendicular  to  the  vertical  axis.  If  the 
error  is  only  just  perceptible,  it  will  not  probably  cause  any 
error  in  the  work. 


EXPLANATOKY  NOTE  ON  THE  USE  OF  THE  TABLES. 


The  logarithms  here  given  are  "  five-place,"  but  the  last 
figure  sometimes  has  a  special  mark  over  it  (e.g.,  g)  which  in- 
dicates that  one-half  a  unit  in  the  last  place  should  be  added. 
For  example : 


the  value 
.69586 
.69586 


includes  all  values  between 
.6958575000  +  and    .6958624999  . 
.6958625000  +  and     .6958674999  . 


The  maximum  error  in  any  one  value  therefore  does  not  ex- 
ceed one-quarter  of  a  fifth-place  unit. 

When  adding  or  subtracting  such  logarithms  allow  a  half-unit 
for  such  a  sign.  For  example : 


.69586  .69586 

,10841  .10841  .1084i 

.12947  .12947  .12947 


.93374  .93375  .9337§ 

All  other  logarithmic  operations  are  performed  as  usual  "and 
are  supposed  to  be  understood  by  the  student. 

313 


TABLE   I.— RADII    OF   CURVES. 


Deg. 

0° 

1° 

2° 

3° 

Deg. 

Miu. 

Radius. 

Log  It 

Radius. 

Log  a 

Radius. 

Log  It 

Radius. 

Log  It 

Min. 

O 

I 

2 

3 
4 
5 

343775 
171887 
114592 
85944 
68755 

oo 

5.5362? 
5.23524 

5-°59I5 
4.9342? 
4.83736 

5729.6 
5635.7 

5544-8 
5456.8 
5371.6 
5288.9 

3.758i3 
.75095 
.74389 
.73694 
.73010 

.72336 

2864.9 
2841.3 
2818.0 
2795.1 

2772.5 
2750,4 

3-457II 
•45351 

-44993 
.44639 
.44287 

•43939 

^1910.1 
1899.5 
1889.1 
1878.8 
1868.6 
1858.5 

3.28105 
.27864 
.27625 
.2738? 
.27151 
.269l5 

0 
I 

2 

3 
4 
5 

6 

8 
9 

10 

57296 
49111 
42972 
38i97 
34377 

4.75812 
.6911? 

.63318 
.  58203 
.5362? 

5208.8 
5131.0 
5055.6 
4982.3 
4911.2 

3.71673 
.71026 

.70377 
.69743 
.69113 

2728.5 
2707.0 
2685.9 
2665.1 
2644  .  6 

3-43593 
.43249 
.42909 
.42571 
.42235 

1848.5 
1838.6 
1828.8 
1819.1 
1809.6 

3.26681 
.26443 
.26217 
.25985 

•25757 

6 

8 
9 

10 

ii 

12 
13 

H 

15 

31252 
28648 
26444 

24555 
22918 

4.49488 

.45709 
.42233 
.39014: 
.36013 

4842.0 
4774-7 
4709.3 
4645.7 
4583.8 

3.68502 

.67895 
.67296 
.66705 
.66122 

2624.4 
2604.5 
2584.9 
2565.6 
2546.6 

3.41903 
.41572 
.41245 
.40919 

.40597 

iSoo.I 
1790.7 

I78I.5 
1772.3 
1763.2 

3.25529 
.25303 
.2507? 
.24853 
.  24629 

ii 

1  2 
13 
14 
15 

16 

17 
18 

19 

20 

21486 

20222 
19099 
18093 
17189 

4.33215 
.30582 
.28100 

.25752 
.23524 

4523.4 
4464.7 
4407.5 
435L7 
4297  -  3 

3.65547 
.64979 
.64419 
.63865 
.63319 

2527.9 
2509.5 

249L3 

2473.4 
2455.7 

3-40275 
.39958 
.39642 

.39329 
•3901? 

1754.2 
1745-3 
1736.5 
1727.8 
I7I9.I 

3.2440? 
.24185 
.23967 
.23748 
•23530 

16 

17 
18 

*9 

20 

21 

22 

23 

24 

25 

16370 
15626 
H947 
14324 

I3751 

4.2140$ 
-19385 
.17454 
.15605 
.13833 

4244.2 

4192.5 
4142.0 
4092.7 

4044.5 

3.62780 
.62247 
.61726 
.61206 
.60685 

2438.3 
2421.  i 

2404  .  2 

2387.5 
2371.0 

3-38703 
.38401 
.38097 
.37794 
.  37494 

1710.6 
I702.I 
1693.7 

1685.4 
1677.2 

3-233H 
.23093 
.22884 
.22676 

.22453 

21 

22 
23 
24 
25 

26 
27 
28 
29 
30 

13222 
12732 
12278 
11854 
11459 

4.12130 
.10491 
.08911 
.0738? 
.05915 

3997.5 
395L5 
3906.6 
3862.7 
3819.8 

3.60173 

.59676 
.59186 
.58689 
.  58204 

2354-8 
2338.8 
2323.0 
2307.4 
2292.0 

3-37I95 
.36899 
.  36604 
.36312 
.36021 

I  669  .  I 

1661.0 
1653.0 
1645.1 
1637.3 

3.22247 
.22037 
.2182? 
.21619 
.21412 

26 
27 
28 

29 

30 

31 

32 

33 
34 
35 

11090 

10743 
10417 

IOIII 

9822.2 

4.04491 
.03112 
.01776 
4.00479 
3.99221 

3777-9 
3736.8 
3696.6 

3657.3 
3618.8 

3.57724 
.57250 
.56786 
.56316 
.55856 

2276.8 
226l  .9 
2247  .  I 
2232.5 
22l8.  I 

3-35733 
•35446 
.35162 

.34879 
.34598 

1629.5 
1621.8 
1614.2 
1606.7 

1599-2 

3.21  206 

.21000 
.20796 
.20593 
.20396 

31 

32 

33 

34 
35 

36 
37 
38 

39 

40 

9549-3 
9291.3 
9046.7 
8814.8 
8594.4 

3-9799? 
.9680? 

.95649 
.94521 
.93421 

358i.i 
3544.2 
35o8.o 
3472.6 
3437.9 

3-55401 
.54951 
.  54506 
•  54o6§ 
•53629 

2203.9 
2189.8 
2176.0 
2162.3 
2148.8 

3.34318 
.34041 
.33765 
.3349? 
.33219 

1591.8 

1584.5 
1577-2 
1570.0 
1562.9 

3.20189 
.19988 
.19789 
.19590 
.19392 

36 

% 

39 
40 

4i 
42 
43 
44 
45 

8384.8 
8185.2 
7994-8 
7813.1 
7639.5 

3-92349 
.91302 
.90281 
.89282 
-88305 

3403.8 
3370.5 
3337-7 
3305.7 

3274.2 

3.53197 
.52769 
.52345 
.51925 
.51510 

2135.4 
2122.3 
2109.2 
2096.4 
2083.7 

3.32949 
.32680 
.32412 

.32147 
-31883 

1555-8 
i  548  .  8 

1541-9 
1535.0 
1528.2 

3.I9I95 
.18999 
.18804 
.I86l6 
.18417 

4i 

42 

43 
44 
45 

46 

47 
48 

49 
50 

7473.4  3.87352 
7314.4   .86418 
7162.0   .85503 
7015.9   .84608 
6875-6   .83731 

3243.3 
3213.0 

3183.2 
3154.0 
3125.4 

3.51093 
.50691 
.50287 
.49885 
.49490 

207  I  .  I 
2058.7 

2046  .  5 
2034.4 
2022.4 

3.31621 
.31360 
.31101 
.30843 
.30587 

1521.4 

1514.7 
i  508  .  i 
1501.5 
U95.  o 

3.18224 
.18032 
.17842 
.17652 
.17462 

46 

47 
48 

49 
50 

5i 
52 
53 
54 

55 

6740.7 
6611.1 
6486  .  4 
6366.3 
6250.5 

3.82871 
.8202? 
.81200 
.80383 
.7959? 

3097.2 
3069.6 
3042.4 
3015.7 
2989.5 

3.49097 
.4870? 
.48321 
•47939 
•47559 

2OIO.6 

1998.9 
1987.3 
1975.9 

1964.6 

3-30332 
.30079 
.2982? 
.29577 
.29323 

1488.5 
1482.1 

1475-7 
1469.4 
1463.2 

3.17274 
.17087 
.16900 
.16714 
.16529 

5i 

52 
53 
54 

55 

56 
57 
58 

I9 
60 

6138.9 
6031.2 
5927.2 
5826.8 

5729.6 

3.78809 
.  78046 
.77285 
.76542 
•75813 

2963.7 
2938.4 

2913.5 
2889.0 
2864.9 

3-47183 
.46811 
.46441 
.46075 
-457II 

1953.5 
1942.4 

1931-5 

1920.7 

1910.1 

3.29081 
.28835 
.28590 
.28347 
.28105 

H57.0 
1450.8 
1444-7 
1438.7 

1432.7 

3.  *  6344 
.l6l6l 

.15978 
.15796 
.15615 

56 
57 
58 

59 
60 

314 


TABLE    I.— RADII    OF    CURVES. 


Deff. 

4° 

5° 

6° 

7° 

I>e&. 

Miu. 

Kadi  us. 

Lo*  -B 

Radius.     Lo*  /.' 

Had  i  us. 

Log  K 

UadiuN. 

Log  Ji 

Mil.. 

0 
I 

2 

3 
4 
5 

1432.7 
1426.7 
1420.8 
I4I5.0 
1409.2 
I403-5 

3.I56I5 
•15434 
.15255 
.  I  5076 

.1489? 
.14720 

1146.3 
1142.5 
1138.7 

H34.9 
II3I  .2 
II27.5 

3.05929 

.05784 
.05646 

.05497 
•05354 
.05211 

955-37 
952.72 
950.09 
947.48 
944.88 
942  .  29 

2.98017 

.97896 

•97776 
.97657 
•97537 
•97418 

819.02 
817.08 
815.14 
813.22 
8II.30 
809  .  40 

2.91329 

.91226 

.91123 

.91021 

.90918 
.90816 

0 
I 

2 

3 
4 
5 

6 

8 
9 

10 

1397-8 
I392.I 
1386.5 
1380.9 
1375-4 

3*.  1  4543 
'  .14367 
.14191 
.14017 
.13843 

II23.8 
I  1  20  .  2 
III6.5 
III2.9 
1109.3 

3.05069 
.04928 

.04787 
.04646 
.04506 

939.72 
937.16 
934.62 
932-09 
929.57 

2.97300 
.97181 
.97063 

.96945 
.96828 

807  .  50 
805.61 

803.73 
801.86 
8CO.OO 

2.90714 

.90612 

.90511 

.90410 

.00309 

6 

8 
9 

10 

ii 

12 
13 
H 
15 

1369.9 
1364.5 
I359-I 
1353-8 
1348.4 

3.13669 
.13497 
••13325 
.13154 
.12983 

H05.8 
IIO2.2 
1098.7 
1095.2 
1091.7 

3.04366 
.04227 
.04088 

•03949 
.03811 

927.07 
924.58 
922.  10 
919.64 
917.19 

2.96711 
.96594 
.96478 
.9636? 
.  96246 

798.14 
796.30 
794.46 
792.63 
790.81 

2  .  90208 
.0010? 
.9000? 
.8990? 
.89807 

ii 

12 
13 
14 
15 

16 

17 
18 

19 

20 

1343-2 
1338.0 
1332.8 
1327.6 
1322.5 

3.12813 
.12644 
.12475 
.12307 
.12146 

1088.3 
1084.8 

1081  .4 
1078.1 
1074.7 

3.03674 
.03537 
.03400 
.03264 
.03128 

9H-75 
912.33 
909.92 
907.52 
905-13 

2.96136 
.96015 
.95900 
•95785 
.95671 

789.00 
787.20 
785.41 
783.62 
781.84 

2  .  89708 
.89608 
.89509 
.89416 
.89312 

16 

17 

18 
19 

20 

21 

22 

23 
24 

25 

I3I7.5 
I3I2.4 

1307.4 
1302.5 
1297.6 

3-II974 
.11808 
.11642 
.11477 
.11313 

1071.3 
1068.0 
1064.7 
1061.4 
1058.2 

3.02992 
.02857 
.02723 
.02589 
.02455 

902.76 
900.40 
898.05 
895.71 
893.39 

2-95557 
•95443 
.95330 
.95217 
.95104 

780.07 
778.31 
776.55 
774.81 
773.07 

2.89213 
.89115 
.89017 
.88919 
.88821 

21 

22 
23 
24 

25 

26 
27 
28 
29 
30 

1292.7 
1287.9 
1283.1 

1278.3 
1273.6 

3.11150 
.  10987 
.10825 
.10663 
.  10502 

1054.9 
1051.7 
1048.5 

1045-3 
1042  .  i 

3.02322 
.02189 
.02055 
.01924 
.01792 

891.08 
888.78 
886.49 
884.21 
881.95 

2.94991 
.94879 
.94767 
.94655 
.94544 

77L34 
769.61 
767.90 
766.19 
764.49 

2.88724 
.88627 
.88536 
.88433 
.88337 

26 
27 
28 

29 
30 

31 

32 

33 

34 
35 

1268.9 
1264.2 
1259.6 
1255.0 
1250.4 

3-10341 
.10182 

.10022 
.09864 
.09705 

1039.0 

1035-9 
1032.8 
1029.7 
1026.6 

3.01661 
.01536 
.01400 
.01270 
.01140 

879-69 
877-45 
875.22 
873.00 
870.80 

2-94433 
.94322 
.94212 
.94101 
.9399? 

762.80 
76l.II 

759-43 
757.76 
756.10 

2.88241 
.88145 
.  88049 

.87953 
.87858 

31 
32 

33 
34 

35 

36 
37 
38 

39 
40 

1245.9 
1241.4 
1236.9 
1232.5 
I228.I 

3.09548 
.09391 
.09234 
.00079 
.08923 

1023.5 
1020.5 
1017.5 
1014.5 
ion  .5 

3.01010 
.00882 
.00753 
.00625 
.00497 

868.60 
866.41 
864  .  24 
862.07 
859.92 

2.93882 
.93772 
.9366§ 
•93554 
.93446 

754-44 
752.80 
75I.I6 
749-52 
747.89 

2.87762 
.87668 
.87573 

.87478 
.87384 

36 
37 
38 
39 
40 

4i 
42 
43 
44 
45 

1223.7 
1219.4 
1215.  I 
I2IO.8 

i  206  .  6 

3.08769 
.08614 
.08461 
.08308 
.08155 

1008.6 
1005.6 
1002.7 

999.76 
996.87 

3.00370 
.00242 
3.00116 

2.9998§ 
.99863 

857.78 
855.65 

853.53 
851.42 
849.32 

2.9333? 
.93229 
.93122 
.93014 
.92907 

746.27 
744-66 
743.06 
741.46 

739-86 

2.87290 
.87196 
.87102 
.87003 
.86915 

4i 
42 
43 
44 
45 

46 

47 
48 

49 
50 

1202.4 
1198.2 
1194.0 
1189.9 
1185.8 

3.08003 
.07852 
.07701 
.07550 
.07400 

993-99 
99LI3 
988.28 

985.45 
982.64 

2.99738 
.99613 
.99488 
.99363 
.99239 

847-23 
845.15 
843.08 

841  .02 
838.97 

2.92800 
.92693 
•92587 
.92486 

.92374 

738.28 
736.70 

735-J3 
733-56 
732.oi 

2.86822 
.86729 
.86636 
.86544 
.8645? 

46 
47 
48 

49 
5o 

51 

52 
53 
.54 
55 

1181.7 
1177.7 
1173.6 
1169.7 
1165.7 

3.07251 
.07102 
•06954 
.06806 
.06653 

979-84 
977.o6 
974-29 

971-54 
968.81 

2.99115 
.98992 
.98869 

.98746 
.98624 

836.93 
834.90 
832.89 
830.88 
828.88 

2.92269 
.92163 
.92053 

•91953 
.91849 

730.45 
728.91 
727.37 
725.84 
724-31 

2.86359 
.8626? 

.86175 
.86084 

•85992 

5i 

52 
53 
54 
55 

56 

H 

59 
60 

1  161.8 

1157.9 

1154.0 
1150.1 
[146,3 

3.06511 
.06365 

.  062  i  9 
.06074 
.05929 

966.09 

963-39 
960.70 
958.03 
955-37 

2.98501 
.98380 

.98258 
.98137 
.98017 

826.89 
824.91 
822.93 
820.97 
819.02 

2.91744 
.91646 
.9^536 
.9H33 
.91329 

722.79 
721.28 

719-77 
718.27 
716.78 

2.85901 
.85816 
.85719 
.85629 
•85538 

56 
57 
58 
59 
60 

315 


TABLE    L— RADII    OF    CURVES. 


Deg. 

8 

o 

9 

o 

1 

)° 

11 

° 

l>«ir. 

Min. 

Radius. 

Los  K 

Radius. 

Log  M 

Radius. 

Log  K 

Uuuiu*. 

i,.,*  i. 

Min. 

O 
I 

2 

3 

4 

5 

716.78 
715.29 
713.81 
712-34 
710.87 
709.40 

2-85538 

.85448 
.85358 
.85263 

•85i78 
.85089 

637.27 
636.  10 

634.93 
633.76 

632.60 

631.44 

2.80432 
.80352 
.80272 
.80192 
.80113 
.80033 

573.69 
572.73 
57L78 
570.84 
569.90 
568.96 

2.7586? 

.75795 
.75723 
.75651 
•75579 
.75508 

521.67 
'520.88 
520.  10 

519.32 
518.54 
5^.76 

2.71739 
.71674 
.71603 
.71543 
.71478 
.71413 

0 

I 
2 

3 

4 
5 

6 

8 
9 

10 

707.95 
706.49 
705.05 
703.61 
702.17 

2.85000 
.84911 
.84822 

.84733 
.84644 

630.29 
629.14 

627.99 
626.85 
625.71 

2.79954 
.79874 
•79795 
•79716 
.79637 

568.02 

567.09 
566.16 

565-23 
564.31 

2.75436 
.75365 
.75293 

.75222 

•  .75i5i 

516.99 

516.21 

515.44 
514.68 

513.91 

2.71348 
.71283 
.71218 

.71153 
.71083 

6 

7 
8 

9 

10 

II 

12 
13 
H 
15 

700.75 

699.33 
697.91 
696  .  50 
695.09 

2.84556 
.84468 
.84380 
.84292 
.  84204 

624.58 
623.45 

622.32 

621  .20 
62O  .  09 

2.79558 
.79480 

.79401 
.79323 
.79245 

563.38 
562.47 
561.55 
560.64 

559-73 

2.75086 
.75009 

•74939 
.74868 
.74798 

513.15 
512.38 
511.63 
510.87 

510.  1  1 

2.71024 

.70959 
-70895 
.70831 

.70767 

ii 

12 
13 
H 
15 

16 

17 
18 

19 

20 

693.70 
692.30 
690.91 

689.53 
688.16 

2.84117 
.84029- 
.83942 
.83855 
.83768 

618.97 
617.87 
616.76 
615.66 
614.56 

2.79167 
.79089 
.79011 
.78934 
.78855 

558.82 
557.92 

557-02 
556.12 

555-23 

2.74727 
.74657 
.74587 
.74517 

-74447 

509.36 
508.61 
507.86 
507.12 
1:06.38 

2.70702 
.70633 

.70575 
.70511 

.7044? 

16 

17 
18 

!9 

20 

21 
22 
23 
24 

25 

686.78 
685.42 
684.06 
682.70 
681.35 

2.83682 
.8359$ 
.83509 
.83423 

.833.37 

613.47 
612.38 

6  i  i  .30 
610.21 
609.14 

2.78779 
.78702 
.78625 
.78548 
.78471 

554-34 
553.45 
552.56 
551.68 
550.80 

2-74377 
.74307 
.74238 
•74»68 
.74099 

505.64 
504.90 

504.16 

503.42 

502.69 

2.70383 
.70320 
.70257 
.70193 
.70130 

21 

22 
23 
24 

25 

26 

27 
28 
29 
30 

680.01 
678.67 
677.34 
676.01 
674.69 

2.8325! 
.83166 
.83086 

.8299? 
.82910 

608.06 
606  .  99 

605.93 
604.86 
603  .  80 

2.78395 
.78318 
.78242 
.78165 
.78089 

549.92 
549.05 
548.17 
547.30 
546  .  44 

2  .  74030 
.73961 
.73892 
.73823 
•73754 

501.96 
501.23 
500.51 
499.78 

499  -  °6 

2  .  70067 
.70004 
.69941 
.69878 
.69815 

26 
27 
28 
29 
30 

31 
32 

33 
34 

35 

673.37 
672.06 

670.75 
669.45 
668.15 

2.8282^ 
.82746 
.82656 
.82571 

.82487 

602.75 
601  .70 
600.65 
599.61 
598.57 

2.78013 
.77938 
.77862 

.77785 
•777II 

545-57 
544.71 
543-86 
543-00 
542.15 

2.73685 
.73617 

•73548 
.7348o 
.73412 

498.34 
497.62 
496.91 
496.19 
495.48 

2.69752 
.  69690 
.69627 
.69565 
.69503 

31 

32 

33 
34 
35 

36 
37 
38 

39 

40 

666.86 
665.57 
664.29 
663.01 
661.74 

2.82403 
.82319 
.82235 
.82152 
.82063 

597-53 
596.50 
595-47 
594-44 
593-42 

2.77636 
.7756i 
.77486 
•774II 
.77336 

54L30 
540.45 
539.6i 
538.76 
537.92 

2.73343 
.73275 
.7320? 
.73140 
.73072 

494-77 
494.07 
493.36 
492.66 

49  i  -96 

2.69446 

.69378 
•69316 
.69254 
.69192 

36 

% 

39 
40 

4i 
42 

43 
44- 
45 

660  .  47 
659.21 

657.95 
656.69 

655.45 

2.81985 
.  8  i  902 
.81819 

.81736 
.81653 

592.40 
59I-38 
590.37 
589-36 
588.36 

2.7726! 
.77187 
.77112 
•77038 
.  76964 

537.09 
536.25 
535-42 
534.59 
533-77 

2.73004 

.72937 
.72869 
.72802 
•72735 

491  .26 
490.56 
489.86 
489.17 
488.48 

2.69131 
.  69069 
.6900^ 
.68946 
.68884 

4i 
42 
43 
44 
45 

46 

47 
48 

49 
50 

654.20 
652.96 

65L73 
650.50 

649.27 

2.81571 
.81489 
.81405 
.81324 
.81243 

587-36 
586.36 
585-36 
584.37 
583-38 

2.76890 
.76815 
.76742 
.76669 
.76595 

532.94 
532.12 

531.3° 
530.49 
529.67 

2.72668 
.72601 
.72534 
.72467 
.72401 

487.79 
487.10 
486.42 

485.73 
485.05 

2.68823 
.68762 
.68701 
.68640 

.68579 

46 
47 
48 
49 
50 

51 

52 
53 
54 
55 

648.05 
646.84 
645.63 
644.42 
643.22 

2.81161 
.81079 
.  80998 
.80917 
.80836 

582.40 
581.42 
580.44 
579-47 
578.49 

2.76522 

.76449 
.76376 
.76303 
•  76230 

528.86 
528.05 
527.25 
526.44 
525.64 

2.72334 
.7226? 
.72201 

.72135 
.72069 

484-37 
483-69 
483.02 

482.34 
481.67 

2.68518 

.68457 
.68395 

.68335 
.68275 

5i 

52 
53 
54 
55 

56 
57 
58 

59 
60 

642  .  02 
640.83 
639.64 

638.45 
637.27 

2.80755 

.80674 
.80593 
.80513 
.80432 

577-53 
576.56 
575-60 
574-64 
573-69 

2.76157 
.76084 
.76012 

•75939 
.7586? 

524.84 
524.05 

523-25 
522.46 
521.67 

2.72003 

.71937 
.71871 
.71805 
.71739 

481.  oo 
480.33 
479.67 
479.0° 
478.34 

2.68214 
.68154 
.  68094 
.68033 
.67973 

56 

57 
58 
59 
60 

316 


TABLE    I.— RADII    OF    CURVES. 


Dec. 

12° 

2 

4 
6 
8 

Radius. 

Log  K 

Deg.  j  Radius.   Log  R 

I>eg. 

Radius. 

Log  R 

Deg.  Radius.    Log  it 

478.34 

477-02 

475-71 
474.40 

473-  I0 

2.67973 
.67853 
.67734 
.67614 
•6749§ 

14° 

2 

6 
8 

410.28  2.61307 
409.31   .61205 
408.34  .61102 
407.38  j  .61000 
406.42  i  .60898 

16° 

5 

10 

15 

20 

25 

359-26 
357-42 

355-59 
353-77 
35L98 
350.21 

2-55541 
•55317 
.55094 
.54872 
.54652 
•54432 

21° 

10 
20 

30 

40 

50 

274-37 
272.23 
270.13 
268.06 
266.02 
264.02 

2.43833 

•43494 
•4315? 
.42823 
.42492 
-42163 

10 
12 

u 

16 
18 
20 

22 
24 
26 
28 

471.81 

470.53 
469.25 

467-98 
466.72 

2.67375 
.67253 
.67146 
.67022 
.66905 

10 
12 
H 

16 
18 

405.47  2.60796 
404.53,  .6069^ 
403.581  .60593 
402.65   .60492 
401  .71   .60391 

30 

35 
40 

45 
50 
55 

348.45 
346.71 
344-99 
343-29 
341.60 

339-93 

2.54214 

i  -53997 
!  -53786 
•53565 
-53351 
•53138 

22° 

10 
20 
30 
40 
50 

262.04  2.41837 
260.10   .41513 
258.18   .41192 
256.29   .40873 

254.43   .40557 
252.  60  '  .40243 

465-46 
464.21 
462.97 

461.73 
460.50 

2.66788 
.66671 
.66555 
.66439 
.66323 

20 

22 

24 
26 

28 

400.78  2.60291 
399.86   .60196 
398.94,  .60096 
398.02   .59996 
397-H  ,  .59891 

17° 

5 

10 

'5 

20 
25 

338-27 
336-64 
335-01 
333-41 
331-82 
330.24 

2.52927 
.52716 
.52506 
.52297 
i  .52090 
1  -51883 

23° 

10 
20 
30 
40 
50 

250.79 
249.01 
247  .  26 

245-53 
243-82 
,  242.14 

2.39931 
.39622 

•39315 
.39016 
.  38707 
-  38407 

30 
32 

34 

II 

3b 

459.28 
458.06 
456.85 
455-65 
454-45 

2.66207 
.66092 

.65977 
.65863 
.65748 

30 
32 

34 
36 
38 

396.20  2.59791 
395.30   .59692 
394.40   .59593 

393  •  5°   •  59494 
392.61   .59396 

3° 
35 
40 
45 
50 
55 

328.68 

327-13 
325.60 

324-09 
322.59 
321  .  10 

2.51677 
!  •  5H72 
.51269 
.51066 
.  50864 
.  50663 

24°  240.49 
.10  238.85 

20   237.24 

30  235.65 
40  234.08 
50  232.54 

2.38109 
.37813 
•37519 
•37227 

•36937 
•36649 

40 
42 
44 
46 
48 
50 
52 
54 

^8 

5b 

453-26 
452.07 
450.89 
449-72 
448.56 
447-40 
446-24 
445-09 
443-95 
442.81 

2.65634 
.65521 
.65407 
.65294 
.65181 

40 

42 
44 
46 
48 

391.72  12.59298 
390.84  .59199 
389-96  !  .59102 
389  .  08  j  .  59004 
388.21  j  .58907 

18° 
5 

10 

15 

20 
25 

319.62 
318.16 
316.71 
315.28 
313.86 
312.45 

2  .  50464 
.50265 
.50067 
.  49869 
•49673 

•49478 

25°   231.01 
30  226.55 
20°   222.27 
30  !  218.15 

2.36363 
•3551? 
.34688 

•3387S 

2.65069 
.64957 
.64845 
.64733 
.64622 

50 
52 
54 
56 
58 

387.34  2.58809 
386.48  .58713 
385.62   .58616 
384.77   .58519 
383.91   .58423 

27   214.18  2.33073 
30  210.36   .32295 
28°  206.68   .31529 
30  j  203.13   .30776 

13J 

6 

8 

10 

I  ""* 

U 
16- 
18 

20 

22 

24 
26 
28 

30 
32 

£ 

38 

441.68 
440.56 
439-44 
438.33 
437.22 

436.12 
435-02 

433-93 
432-84 
43L76 
430.69 
429.62 
428.56 
427-50 
426.44 
425.40 
424-35 
423-32 
422.28 
421  .26 

2.64511 
.64400 
.64290 
.64180 
.  64070 

15° 

2 

4 
6 
8 

383.06  2.58327 
382.22  .58231 
381.38  .58135 
380.54  .58046 
379.71   -57945 

30   311.06 

35  !  309-67 
40  308.3° 
45  306.95 
50  305  .  60 
55  !  304-27 

2.49284 
.49096 
.48898 
.48706 

.48515 
.48325 

29   I99-70 
30  196-38 
30°   193.19 

30   i  go  .  09 

2.30037 
.29316 
.28597 

2  .  63966 
.63851 
.63742 
•63633 
.63524 

10 

12 

U 

16 

18 

378.88  2.57850 
378.05  .57755 
377-23  -5/661 
376.41   .57565 
375.60  .57472 

lj>°  302.94 
5  301.63 
10  300.33 
15  299.04 

20   297.77 
25   296.50 

2.48136 
•4/948 
.47766 

•47573 
.47388 

!  .47203 

31   187.10 
32   181.40 
33   176.0; 
34   171-02 
35   166.28 

2.27207 
.25863 

•24563 
-23303 
.22083 

2-63416 
.63308 
.63201 
.63093 
•62985 

20 

22 

24 
26 
28 

374.79  2.57378 
373-98|  -57284 
373-171  .57191 
372.37  .57097 
371.57  .57004 

30 
37 
38 
39 
40 

161.80 

I57o8 
I53-58 

'149-79 
146.19 

2  .  20899 

•19749 
•18633 

•17547 
.16492 

30   295.25 

35  294.00 
40  292.77 
4?  291.55 
50  290.33 
55  289-i3 

2-47018 
•46835 
.46655 
.46471 
.46289 
.46109 

2.62879 
.62773 
.62665 
.62566 
.62454 

30 

32 

3 

38 

370.78  2.56911 

369-99  -56819 
369.20  .56725 
368.42  .56634 
367.64;  .56542 

41 
42 
43 
44 
45 

142.77 
I39-52 
136-43 
133-47 
i  30  .  66 

2.15464 
.14464 
.13489 
•12539 
-  H6l3 

-0   287.94 
5  286.76 
10  285.58 
15  284.42 

20   283.27 
2;   282.  12 

2.45930 
•45751 
•45573 
.45396 
.45219 
.45044 

40 
42 

44 

46 
48 

420.23 
419.22 
418.20 

417-19 
416.19 

2.62349 
.62243 
.62138 
.62034 
.61929 

40 
42 

44 
46 
48 

366.86  2.  56450 
366.09   .56358 
365.31  I  .56265 

364.55  !  .56175 
363.78  .56084 

4(5 
47 

48 
49 
50 

127.97 

125-39 
122.93 
120.57 
118.31 

2.  10709 
.09827 
.08965 
.  08  T  24 
.07302 

30 

35 
40 

45 
50 
55 

280.99 
279.86 
278.75 
277-64 
276.54 
275-45 

2.44869 
.44694 
•4452f 
•4434^ 
.44176 
.44004 

So 

52 

% 

58 
14° 

415.19 
414.  20 
413.21 
412.23 
411.25 

2.61825 
.61721 
.61617 
.61514 
.61416 

50 

52 
54 
56 
58 

363.02  2.55993 
362.26;  .55902 
361.51  |  .55812 
360.76;  .55721 
^60  01   s^6^T 

52 
U 
50 

58 

114.06 
110.13 
106.50 
103.13 

IOO.OO 

2.05713 
.04192 
.02736 
.01340 
2.00000 

21°  274.37 

2-43833 

410.28 

2.61307 

10°   359-26  2.55541 

60 

317 


TABLE   II.— TANGENTS,  EXTERNAL   DISTANCES,  AND  LONG   CHORDS    FOR   A 

1°  CURVE. 


A 

Tangent 
T. 

Ext.Dist. 
E. 

LoiigCh'd 

£c. 

A 

Tan  ire  nt 
T. 

Ext.Dist. 

jr. 

LonsrCli  (1 
LC. 

A 

Tangent 
f. 

Ext.Dist. 

E. 

LoiigCh'd 
LC. 

1° 

10? 

20 

30 
40 

50 

50.00 

58.34 
66.67 

75.01 
83.34 
91.68 

0.218 
0.297 
0.388 
0.491 
0.6o6 

0.733 

I  OO  .  OO 

116.67 
133-33 

150.00 

166.66 
183.33 

11° 

10 

20 
30 
40 
50 

551.70 
560.  II 
568.53 

576.95 
585.36 

593-79 

26.500 

27-313 
28.137 
28.974 
29.824 
30.686 

i  098  .  3 
1114.9 

H3I.5 

1148.  i 
1164.7 
1181.2 

21° 

10 
20 
30 
40 
50 

1061  .9 
1070.6 
1079.2 
1087.8 
1096.4 
1105.  i 

97.58 

99.15 
100.75 
102.35 
103.97 
105.60 

2088.3; 
2104.71 
2121  .  I 
2137.4! 
2153-8! 
2170.2 

2° 

10 
20 

30 
40 

50 

IOO.OI 

108.35 
116.68 
125.02 
133.36 
141.70 

0.873 
.024 
.188 
.364 

.552 
.752 

199.99 
216.66 
233-32 
249.98 
266.65 
283.31 

12° 

10 

20 
30 
40 
50 

602  .  2  1 
610.64 
619.07 
627.50 
635.93 
644-37 

31.561 
32.447 
33-347 
34-259 
35.183 
36.120 

1197.8 
1214.4 
1231  .0 
1247.5 
i  264  .  i 
1280.7 

22° 

10 

20 
30 
40 
50 

1113.7 
1122.4 
1131.0 

II39-7 
1148.4 
1157.0 

107.24 
108.90 
110.57 
112.25 
113-95 

i  i  5  .  66 

2186.  5  i 
2202.9 
2219.2 
2235.6 
2251.9 
2268  .  3 

3° 

10 

20 

30 
40 

50 

1  50  .  04 

158.38 
166.72 
175.06 

I  83  .  40 

191.74 

1.964 
2.188 
2.425 
2.674 

2-934 
3.207 

299-97 
316.63 

333.29 

349-95 
366.61 

383-27 

13° 

10 
20 
30 
40 
50 

652.81 
661.25 
669.70 
678.15 
686.60 
695.06 

37.069 
38.031 
39.006 

39-993 
40.992 
42  .  004 

1297.2 
1313.8 

1330.3 
1346.9 

1363-4 
1380.0 

23° 

10 

20 
30 
40 
50 

1165.7 
1174.4 
1183.1 
1191.8 
1200.5 

1  209  .  2 

117.38 

119.  12 
120.87 
122.63 
124.41 
126.20 

2284.6 
2301.0 

2317.3 
2333-6 
2349.9 
2366.2 

4° 

10 
20 

30 
40 

50 

200.08 

208.43 
216.77 

225.12 

233-47 

241.81 

3.492 
3-790 
4.099 
4.421 

4-755 
5.  100 

399.92 
416.58 

433-24 
449.89 
466.54 
483.20 

U° 

10 
20 
30 
40 
50 

703.51 
711.97 

720.44 
728.00 

737-37 
745-85 

43.029 
44.066 
45.116 
46.178 
47-253 
48-341 

1396.5 
HI3.I 
1429.6 
1446.2 
1462.7 
1479.2 

24° 

10 
20 
30 
40 
50 

I2I7.9 
1226.6 

I235O 
1244.0 
1252.8 
I26I.5 

128.OO 
129.82 
131.65 

I33-50 
135.36 
137.23 

2382.5 
2398-8 
2415.1 
243!-4 

2447  -  7  ; 
2464  .  o 

5° 

10 
20 

30 

40 

50 

250.16 

258.51 
266.86 
275.21 

283.57 
291.92 

5.459 
5.829 

6.  211 

6.606 
7.013 

7.432 

499-85 
516.50 

533-^5 
549.80 
566.44 
583.09 

15° 

10 
20 
30 
40 
50 

754-32 
762.80 
771.29 

779-77 
788.26 

796.75 

49-441 
50.554 
51.679 
52.818 
53.969 
55-I32 

'495-7 
1512.3 
1528.8 

1545-3 
1561.8 

1578.3 

25° 

10 
20 
30 
40 
50 

1270.2 
1279.0 
1287.7 
1296.5 
1305.3 
I3H.O 

139.11 

141  .01 

142.93 
144.85 
146.79 

148.75 

2480.2 
2496.5 
2512.8 
2529.0 

2545-3! 
2561.5 

6° 

10 

20 

30 
40 

50 

300.28 

308.64 
316.99 
325.35 
333.71 

342.08 

7.863 

8.307 
8.762 
9.230 
9.710 

IO.2O2 

599-73 
616.38 
633.02 
649  .  66 
666.30 
682.94 

10° 

10 
20 
30 
40 
50 

805.25 

813.75 
822.25 
830.76 
839.27 
847.78 

56.309 
57-498 
58.699 

59-9J4 
61.  141 
62.381 

1594.8 
1611.3 
1627.8 
1644.3 
1660.8 
1677.3 

26° 

10 
20 
30 

40 

50 

1322.8 
I33L6 
1340.4 
1349-2 
1358.0 
1366.8 

150.71 
152.69 
154.69 
156.70 
158.72 
160.76 

2577-8! 
2594.0! 
2610.3; 
2626.5; 
2642.7! 
2658.9, 

7° 

10 
20 

30 
40 

50 

350.44 
358.81 

367.17 
375.54 
383.91 
392.28 

10.707 
1  1  .  224 

n-753 
12.294 
12.847 
I3-4I3 

699.57 
716.21 
732.84 
749-47 
766.  10 

782.73 

17° 

10 
20 
30 
40 
50 

856  .  30 

864.82 

873-35 
881.88 
890.41 
898.95 

63-634 
64.900 
66.178 
67.470 
68.774 
70.091 

1693.8 
1710.3 
1726.8 
1743-2 
1759-7 
1776.2 

27° 

10 
20 

3° 
40 

50 

1375-6 
1384.4 
1393.2 
1402.0 
I4I0.9 
I4I9.7 

l62.8l 
164.87 
166.95 
I  69  .  04 

I7LI5 

173.27 

2675.1; 
2691.3: 
2707  .  5 
2723-71 

2739-9: 

2756.  i 

8° 

10 
20 

30 
40 

50 

400.66 

409.03 

417.41 
425.79 
434.17 
442.55 

I3-99I 

14.582 
15.184 

15-799 
16.426 
17.066 

799-36 
815.99 
832.61 

849-23 
865.85 
882.47 

18° 

10 
20 
30 
40 
50 

907.49 
916.03 

924.58 
933-13 
941.69 
950.25 

71.421 

72.764 
74.119 
75.488 
76.869 
78.264 

1792.6 
1809.1 
1825.5 
i  842  .  o 

1858.4 
1874.9 

28° 

10 
20 
30 
40 
50 

1428.6 
1437-4 
1446.3 
I455-I 

i  464  .  o 

T472.9 

!75-4i 
177-55 
179.72 
181.89 
184.08 
186.29 

2772.3! 
2788.4 
2804.6 
2820.7 
2836.9 
2853.0; 

9° 

10 
20 

30 
40 

50 

450.93 
459.32 
467.71 

476.  10 

484.49 

492  .  88 

17.717 
18.381 
19.058 
19.746 
20.447 

21.  l6l 

899.09 
915.70 

932-31 
948.92 

965.53 

982  14 

19° 

10 

20 
30 
40 
50 

958.81 
967-38 
975.96 

984-53 
993.12 
1001  .70 

79.671 
8  i  .  092 

82.525 
83.972 

85-431 
86.904 

1891.3 

1907.8 
1924.2 

i  940  .  6 
1957.1 

1973-5 

29° 

10 
20 
30 
40 
50 

I48I.8 
1400.7 
1499.6 
1508.5 
I5I7.4 
1526.3 

188.51 
190.74 
192.99 
195.25 

197-53 
199.82 

2869.  2 

2885.  3  ! 
2901.4 
2917.6; 

2933-7 
2949.8 

10° 

10 

20 

30 
40 

50 

501.28 
509  .  68 
518.08 
526.48 
534.89 
543.29 

21.886 

22  .  624 

23-375 
24.138 
24.913 
25.700 

998.74 
10^5-35 
1031.95 
1048.54 
1065.  14 
1081  .73 

20° 

10 
20 
30 
40 
50 

1010.29 
1018.89 
1027.49 
1036.09 
1044.70 
1053-31 

88.389 
89.888 

91-399 
92.924 
94.462 
96.013 

1989.9 
2006  .  3 
2022.7 
2039.1 

2055-5 
2071.9 

30° 

10 
20 
30 
40 
50 

1535.3 
1544.2 

I553-I 
1562.  I 
I57I.O 
1580.0 

202  .  I  2 
204.44 
206.77 
209  .  1  2 
2  1  I  .  48 

2  i  3  .  86 

2965.9 
2982.0 
2998.1 
3014.2 
3030.2 
3046.3 

11° 

55L70 

26  .  500 

1098.33 

21° 

1061.93 

97-577 

2088  .  3 

31° 

I  589.0 

216.25 

3062.4 

318 


TABLE    II.— TANGENTS,  EXTERNAL    DISTANCES,  AND  LONG   CHORDS    FOR   A 

1°  CURVE. 


A 

Talent 

Kxt.IHst.  LongCh'd 
E.            LC. 

A 

Tan  in-lit 
T. 

Ext.Dist. 
E. 

LongClTd 
LC. 

A     Tai£ent 

Ext.Dist. 
E. 

LongCh'd 
LC. 

31° 

10' 

20 

3° 

40 
50 

1589.0 
1598.0 
1606.9 
1615.9 
1624.9 
1633.9 

216.25 
218.66 
221.08 
223.51 
225.06 
228.42 

3062.4 
3078.4 
3094.5 
3110.5 
3126.6 
3142.6 

41° 

10 
20 
30 
40 
50 

2142.2 
2I5I.7 
2l6l.2 
2170.8 
2180.3 
2189.9 

387.38 
390.71 
394.06 

397-43 
400.82 
404.22 

4013.1 

4028  .  7 
4044.3 
4059-9 
4075-5 
4091.1 

51° 

10 

20 
30 
40 
50 

2732.9 
2743.1 
2753-4 
2763.7 

2773-9 
2784.2 

618.39 
622.81 
627.24 
631.69 
636.16 
640.66 

4933-4 
4948.4 

4963-4 
4978.4 
4993-4 
5008.4 

;j-_> 

10 

20 

30 
40 

50 

1643.0 
1652.0 

1661  .0 
1670.0 
1679.1 
1688.1 

230.90 

233.  '39 
235.90 
238.43 
240.96 

243.52 

3158.6 
3J74-6 
3190.6 
3206.6 
3222.6 
3238.6 

42J 

10 
20 
30 
40 
50 

2199.4 
2209.0 
2218.6 
2228.1 
2237.7 

2247  .  3 

407.64 
411.07 
4H.52 
417.99 
421.48 
424.98 

4106.6 
4122.2 
4137-7 
4I53-3 
4168.8 

4184.3 

52° 

10 
20 
30 
40 
50 

2794-5 
2804.9 
2815.2 
2825.6 

2835-9 
2846.3 

645.17 
649.70 
654.25 
658.83 
663.42 
668.03 

5023-4 
5038-4 
5053-4 
5068.3 

5083-3 
5098.2 

33° 

10 
20 

30 

40 

50 

1697.2 
1706.3 

I7I5.3 
1724.4 

1733-5 
1742.6 

246.08 
248.66 
251.26 

253.87 
256.50 

259-H 

3254.6 
3270.6 
3286.6 
3302.5 
3318.5 
3334-4 

43° 

10 
20 
30 
40 
50 

2257.0 
2266.6 
2276.2 
2285.9 
2295.6 
2305-2 

428.50 
432.04 

435-59 
439-  16 
442.75 
446.35 

4199.8 

42I5-3 
4230.8 
4246.3 
4261.8 
4277.3 

53° 

10 

20 
30 
40 
50 

2856.7 
2867.1 

2877-5 
2888.0 
2898.4 
2908.9 

672.66 

677.32 
681.99 

686.68 
691  .40 
696.13 

5H3.I 
5128.0 
5142.9 
5157.8 
5172.7 
5187.6 

34° 

10 

20 

30 
40 

50 

I75L7 
1760.8 
1770.0 

1779-  i 
1788.2 

1797.4 

261.80 
264.47 
267.16 
269.86 
272.58 
275.31 

3350-4 
3366.3 
3382.2 
3398.2 

34H-  1 
3430-0 

44" 

10 
20 
30 
40 
50 

2314-9 
2324.6 

2334-3 
2344-1 
2353-8 
2363-5 

449.98 
453-62 
457-27 
460.95 
464.64 
468.35 

4292.7 
4308.2 

4323-6 
4339-o 
4354-5 
4369-9 

54° 

10 

20 

3° 

40 

_5o_ 
55° 

10 
20 
30 
40 
50 

2919.4 
2929.9 
2940.4 
2951.0 
2961.5 
2972.1 

700.89 
705.66 
710.46 
715.28 
720.  ii 
724-97 

5202.4 

5217.3 
5232.1 
5246.9 
5261.7 
5276.5 

35° 

10 

20 

30 
40 

50 

1806.  6 

1815-7 
1824.9 
1834.1 

1843-3 
1852.5 

278.05 
280.82 
283.60 
286.39 
289.20 
292.02 

3445-9 
346r.8 

3477-7 
3493-5 
3509.4 
3525-3 

45° 

10 
20 
30 
40 

5° 

2373-3 
2383.1 
2392.8 
2402  .  6 
2412.4 
2422.3 

472.08 
475.82 
479-59 
483.37 
487.16 
490.98 

4385.3 
4400.7 
4416.1 

443L4 
4446.  8 
4462.2 

2982.7 

2993-3 
3003.9 

3014-5 
3025.2 

3035.8 

729-85 
734.76 
739-68 
744.62 
749-59 
754-57 

529L3 
5306.1 
5320.9 
5335-6 
5350.4 
5365.1 

36° 

10 
20 
30 
40 
50 

1861.7 
1870.9 
1880.  i 
1889.4 
1898.6 
1907.9 

294.86 
297.72 
300.59 

303-47 
306.37 

309-29 

354I-I 
3557-0 
3572.8 
3588.6 

3604-5 
3620.3 

46° 

10 
20 
30 
40 
50 

2432.1 
2441.9 
2451.8 
2461.7 
2471.5 
2481.4 

494.82 
498-67 
502.54 
506.42 
510.33 
514.25 

4477-5 
4492  .  8 
4508.2 

4523-5 
4538.8 

4554-1 

56° 

10 
20 
30 
40 
50 

3046.5    759-58 
3057.2    764.61 
3067.91  769.66 
3078.7!  774-73 
3089.4    779-83 
3100.2    784-94 

5379-8 

5394-5 
5409.2 

5423.9 
5438.6 

5453-3 

37° 

10 

20 

30 
40 

50 

1917.1 
1926.4 
1935.7 
1945-0 
1954-3 
1963.6 

312.22 

3I5-I7 
318.13 
321.  ii 
324.11 
327.12 

3636.1 
3651-9 
3667.7 
3683.5 
3699-3 
37i5.o 

47° 

10 
20 
30 
40 
50 

249I-3 
2501.2 

2511  .2 
252I.I 
2531.1 
2541.0 

518.20 
522.16 
526.13 
530.13 
534.15 
538.18 

4569.4 
4584.7 
4599-9 
4615.2 

4630.4 
4645  •  7 

57° 

10 

20 
30 
40 

5° 

3110.9  :  790.08 
3121.7    795.24 
3132.6  I  800.42 
3143.4    805.62 
3154.2    810.85 
3165.1    816.10 

5467-9 
5482.5 
5497-2 
5511.8 
5526.4 
5541.0 

38° 

10 
20 

30 
40 

50 

1972.9 
1982.2 

I99L5 
2000.9 

2010.2 
2019.6 

330.15 
333-19 
336.25 
339-32 
342.41 
345-52 

3730-8 

3746-5 
3762.3 
3778.0 
3793-8 
3809.5 

48° 

10 
20 
30 
40 
50 

2551.0 
2561.0 
2571.0 

2581  .0 

259I.I 
2601  .  I 

542.23 
546.30 
550.39 
554-50 
558.63 

562.77 

4660  .  9 
4676.  i 

4691-3 
4706.5 
4721.7 
4736.9 

58° 

10 
20 
30 
40 
50 

3176.0    821.37 
3186.9    826.66 
3197.8    831.98 
3208.8    837.31 
3219.7;  842.67 
3230.7    848.06 

5555-6 
5570.2 
5584.7 
5599-3 
5613.8 
5628.3 

3r 

10 
20- 
30 
40 

50 

2029.0 
2038.4 

2047  .  8 

2057.2 
2066.6 

2076.0 

348.64 
35I-78 
354.94 
358.11 
361.29 
364-50 

3825.2 

3840.9 
3856.6 

3872.3 
3888.0 
3903.6 

49° 

10 
20 
30 
40 
50 

26ll  .2 
2621  .2 
2631.3 
2641.4 
2651.5 
2661.6 

566.94 
571.12 
575-32 
579-54 
583-78 
588  .  04 

4752.1 

4767-3 
4782.4 

4797-5 
4812.7 
4827.8 

59° 

10 
20 
30 
40 
50 

324L7 
3252.7 
3263.7. 
3274.8 
3285.8 

3296-9 

853-46 
858.89 

864.34 
869.82 

875.32 
880.84 

5642  .  8 

5657.3 
5671.8 
5686.3 
5700.8 
5715.2 

40* 

10 
20 

30 
40 

50 

2085.4 
2094.9 
2104.3 
2113.8 
2123.3 
2132.7 

367-72 

370.95 

374-20 

377-47 
380.76 
384.06 

39I9-3 
3935-0 
3950.6 
3966.3 
398i.9 
3997-5 

50° 

10 
20 
30 
40 
50 

2671.8 
2681.9 
2692  .  I 
2702.3 
2712.5 
2722.7 

592.32 
596.62 
600.93 
605.27 
609  .  62 
614.00 

4842.9 
4858.0 

4873-1 
4888.2 
4903.2 
4918.3 

60° 

10 
20 
30 
40 
90 

3308.0 
33I9.I 
3330-3 
334L4 
3352-6 
3363.8 

886.38 
891.95 
897  .  54 

903-15 
908.79 

9H.45 

5729.7 
5744-1 
5758.5 
5772.9 
5787.3 
58oi.7 
5816.0 

4V     2142.2  i  387.38 

4013.  i 

51° 

2732.9 

618.39! 

4933-4 

>1°      3375-0    920.14 

319 


TABLE    II.— TANGENTS,   EXTERNAL    DISTANCES,  AND  LONG    CHORDS    FOR   A 

1°  CURVE. 


A 

Tangent 
T. 

Ext.Dist. 
E. 

LongCH'd 
LC. 

A 

Tangent 
T. 

Ext.Dist. 
J& 

LongCird 
LC. 

A 

Tangent 
T. 

Ext.Dist. 
E. 

LongCh'd 
LC. 

61° 

10' 

20 
30 
40 

50 

3375-0 
3386.3 
3397-5 
3408.8 
3420.1 
343L4 

920.14 
925.85 
93L58 
937-34 

943-12 
948.92 

5816.0 
5830.4 
5844.7 
5859.1 

5873-4 
5887.7 

71° 

10 
20 
30 
40 
50 

4086  .  9 

4099-5 
4112.1 
4124.8 

4137.4 
4150.1 

1  308  .  2 

I3I5-5 
1322.9 

1330.3 
1337.7 
I345-I 

6654-4 

6668.0 
6681.6 
6695.1 
6708.6 
6722.  i 

81° 
10 

20 
30 
40 
50 

4893.6 
4908.0 
4922.5 

4937-0 

4951-5 
4966  .  i 

1805.3 
1814.7 
1824.  I 
1833-6 
1843.1 
1852.6 

7442  .  2 

7454-9 
7467  -  5 
7480.2 
7492.8 
7505.4 

62° 

10 
20 
30 
40 
50 

3442.7 
3454-1 
3465.4 
3476.8 

3488  .  2 
3499.7 

9!4i5 

960  .  60 
966.48 
972.39 
978.31 
984.27 

5902.0 
5916.3 
5930.5 

5944-8 
5959-0 
5973-3 

72° 

10 

20 
30 
40 
50 

4162.8 
4175.6 
4188.4 

4201  .2 
4214.0 
4226.8 

1352.6 
I  360  .  I 
1367.6 
1375-2 
1382.8 
1390.4 

6735.6 
6749.1 
6762.5 
6776.0 

6789-4 
6802.8 

82° 

10 
20 
30 

40 

50 

4980.7 

4995-4 
5010.0 
5024.8 
5039-5 
5054.3 

1862.2 
1871.8 
1881.5 
1891  .2 
1900.9 
I9I0.7 

7518.0 
7530.5 
7543-1 
7555-6 
7568.2 
758o.7 

63° 

10 

20 
30 
40 
50 

tfll.I 

3522.6 

3534.1 
3545-6 
3557.2 
3568.7 

990.24 
996.24 
1002.3 
1008.3 
1014.4 
1020.5 

5987  •  5 
6001.7 
6015.9 
6030  .  o 
6044.2 
6058.4 

73° 

10 
20 
30 
40 
50 

4239-7 
4252.6 
4265.6 

4278.5 
4291.5 
4304.6 

1398.0 
1405.7 

I413-5 
I42I.2 
1429.0 
1436.8 

6816.3 
6829.6 
6843.0 
6856.4 
6869.7 
6883.1 

83° 

10 
20 
30 
40 
50 

5069  .  2 

5084  .  o 

5099.0 

5U3.9 
5I28.9 

5H3-9 

1920.5 

I930-4 
1940.3 

1950.3 
I  960  .  2 
1970.3 

7593-2 
7605.6 
7618.1 
7630.5 
7643.0 
7655.4 

64° 

10 
20 
30 
40 
50 

3580.3 

359L9 
3603.5 
3615.1 

3626.8 
3638.5 

1026.6 
1032.8 
1039.0 
1045.2 
1051.4 
1057.7 

6072.5 
6086.6 
6100.7 
6114.8 
6128.9 
6143.0 

74:° 

10 
20 

30 
40 
50 

4317.6 
4330.7 

4343-8 
4356.9 
4370.1 
4383.3 

1444.6 

1452.5 
1460.4 
1468.4 
1476.4 
1484.4 

6896.4 
6909.7 
6923.0 

6936  .  2 
6949.5 
6962.8 

84° 
10 

20 
30 
40 
50 

5159.0 
5174.1 
5189-3 
5204.4 
5219.7 
5234.9 

1980.4 
1990.5 
2000  .  6 
2OIO.8 
2021  .  I 
2031.4 

7667.8 
7680.1 

7692.5 
7704.9 
7717.2 
7729.5 

65° 

10 
20 
30 
40 
50 

36|o.2 
3661.9 
3673.7 
3685.4 

3697  •  2 
3709.0 

1063.9 
1070.2 
1076.6 
1082.9 
1089.3 
1095.7 

6157.1 
6171.1 
6185.2 
6199.2 

6213.  2 
6227.2 

75° 

10 
20 
30 
40 
50 

4396.5 
4409.8 

4423-1 
4436.4 
4449-7 
4463  •  i 

1492.4 
1500.5 
1508.6 
I5I6.7 
1524.9 
I533-I 

6976.0 
6989.2 
7002.4 
7015.6 
7028.8 
7041.9 

85° 

10 

20 

30 
40 
50 

5250.3 
5265.6 
5281.0 
5296.4 
5311-9 
5327.4 

2041.7 
2052.  I 

2062  .  5 
2073.0 
2083.5 
2094  .  i 

7741-8 
7754-1 
7766.3 
7778.6 
7790.8 
7803.0 

66° 

10 

20 

3° 
40 

5o 

3720.9 
3732.7 

3744.6 
3756.5 
3768.5 
3780.4 

IIO2.2 

1108.6 
1115.1 
1121.7 
1128.2 
1134.8 

6241  .2 

^l5'2 
6269.  I 

6283.1 
6297.0 
6310.9 

76° 

10 
20 
30 
40 
50 

4476.5 
4489.9 

4503.4 
4516.9 

4530.4 
4544-0 

I54I.4 

1549-7 
1558.0 
1566.3 
1574-7 
I583.I 

7055.0 
7068.2 
7081.3 
7094.4 
7107.5 
7120.5 

80° 

10 
20 
30 
40 
50 

5343-0 
5358.6 
5374-2 

5389-9 
5405  .  6 

5421.4 

2104.7 
2115.3 
2126.0 
2136.7 
2147.5 
2158.4 

7815-2 
7827.4 
7839-6 
7851.7 
7863.8 
7876.0 

07° 

10 
20 
30 
40 

5° 

3792.4 
3804.4 
3816.4 
3828.4 

3840.5 
3852.6 

1141.4 

1148.0 

1154.7 

1161.3 
1168.1 
1174.8 

6324-8 
6338.7 
6352.6 
6366.4 
6380.3 
6394.1 

77° 

10 
20 
30 
40 
50 

4557-6 
4571.2 
4584.8 
4598.5 
4612.2 
4626.0 

I59I.6 
l6oO.  I 

1608.  6 
1617.1 
1625.7 
1634.4 

7I33-6 
7H6.6 
7159.6 
7172.6 
7185.6 
7198.6 

87° 

10 
20 
30 
40 

50 

5437-2 
5453-1 
5469-0 

5484.9 
5500.9 
5517.0 

2  1  69  .  2 
2  I  80  .  2 
2191  .  I 
2202  .  2 
2213.2 
2224.3 

7888.1 
7900.1 
7912.2 
7924.3 
7936.3 
7948  .  3 

68° 

10 
20 
30 
40 
50 

3864.7 
3876.8 
3889.0 
390L2 
3913.4 
3925.6 

1181.6 
1188.4 
1195.2 

1202.0 

1  208  .  9 

I2I5.8 

6408.0 
6421.8 
6435.6 
6449.4 
6463  .  I 
6476.9 

78" 

10 
20 
30 
40 
50 

4639.8 
4653.6 
4667.4 

4681.3 
4695.2 
4709.2 

i  643  .  o 
1651.7 
1660.5 
1669.2 
1678.1 
1686.9 

7211  .6 
7224.5 
7237.4 
7250.4 
7263.3 
7276.1 

88° 

10 
20 

30 

40 

50 

5533-1 
5549-2 
5565.4 
5581.6 
5597-8 
5614.2 

2235-5 
2246.7 
2258.0 
2269.3 
2280.6 
2292.0 

7960.3 
7972-3 
7984.2 
7996.2 
8008  .  i 
8020.0 

G9° 

10 
20 
30 
40 
50 

3937-9 
3950.2 

3962.5 
3974.8 
3987.2 
3999-5 

1222.7 
1229.7 
1236.7 

1243-7 
1250.8 
1257.9 

6490  .  6 
6504.4 
6518.1 
6531.8 

6545-5 
6559.1 

79° 

10 

20 
30 
40 

50 

4723.2 
4737-2 
4751.2 

4765.3 
4779-4 
4793-6 

1695.8 
1704.7 

I7I3-7 
1722.7 

I73I-7 
i  740  .  8 

7289.0 
7301.9 
73I4-7 
7327.5 
7340.3 
7353-1 

89° 

10 
20 
30 
40 
50 

5630.5 
5646.9 

5663-4 
5679-9 
5696.4 

57i3-o 

2303-5 
2315.0 
2326.6 
2338.2 
2349.8 
2361.5 

8031.9 
8043.8 
8055.7 
8067.5 
8079.3 

8091  .  2 

70° 

10 
20 
30 
40 
50 

4011.9 
4024  .  4 
4036  .  8 

4049.3 
4061.8 

4074.4 

1265.0 
I272.I 
1279.3 
1286.5 
1293.7 
1300.9 

6572.8 
6586.4 
6600  .  i 
6613.7 
6627.3 
6640  .  9 

80° 

10 
20 
30 
40 

50 

4808  .  7 
4822.0 
4836.2 
4850.5 
4864.8 
4879.2 

^749-9 
1759.0 
1768.2 

1777-4 
1786.7 
1796.0 

7365-9 
7378.7 
7391-4 
7404.1 
7416.8 
7429.5 

90° 

10 
20 

30 
40 
50 

5729.7 
5746.3 
5763-1 
5779-9 
5796.7 
5813.6 

2373-3 
2385.1 
2397.0 

2408  .  9 
2420  .  9 
2432.9 

8103.0 
8II4.7 
8126.5 
8138.2 
8150.0 
8l6l.7 

71° 

4086  .  9 

1308.2 

6654.4 

81° 

4893.6 

1805.3 

7442  •  2 

91° 

5830.5 

2444  •  9 

8I73-4 

120 


TABLE    III.— SWITCH    LEADS   AND    DISTANCES. 


LEAD-RAILS    CIRCULAR 

THROUGHOUT; 

GAUGE  4'  8|".     See  §  262. 

Frog 
Mumber 

FrogAngle(-F) 

Lead  (L) 
(Eq.  79). 

Chord  (QT) 
(Eq.  77). 

Radius  of  Lead 
Rails  (r,Eq.  78). 

Log  r. 

Degree  of 
Curve  (d). 

Frog 
Number 

4 

14°  15'   oo" 

37 

.67 

37 

38 

ISO- 

67 

2.I780I 

38°    46' 

4 

4-5 

12     40     59 

42 

•37 

42. 

12 

190. 

69 

.  28032 

30     24 

4-5 

5 

ii     25     16 

47 

.08 

46. 

85 

235- 

42 

.37i8§ 

24     32 

5 

5-5 

10      23      20 

51 

•79 

51- 

58 

284. 

85 

.45462 

20      13 

5  5 

6 

9     31     38 

56 

•50 

56 

30 

339- 

00 

.  53020 

16      58 

6 

6-5 

8     47     51 

'     61 

.21 

61 

03 

397- 

85 

.59972 

14     26 

6.5 

7 

8     10     16 

65 

.92 

65- 

75 

461. 

42 

.66409 

12      26 

7 

7-5 

7     37    41 

70 

.62 

70 

47 

529. 

69 

.  72402 

10    50 

7.5 

8 

7    09     10 

75 

•33 

75 

19 

602. 

67 

.78007 

9     31 

8 

8-5 

6    43     59 

80 

.04 

79 

90 

680. 

36 

-83273 

8     26 

8-5 

9 

6    21     35 

84 

•75 

84 

62 

762. 

75 

.88238 

7     31 

9 

9.5 

6    01     32 

89-46      ' 

89. 

33 

849. 

85 

.92934 

6     45 

9-5 

10 

5     43     29 

94 

•17 

94-05 

941.67 

2 

.97389 

6     05 

10 

10.5 

5     27     09 

98 

.87 

98 

76 

1038. 

19 

3 

.01627 

5     32 

10.5 

ii 

5     12     18 

103 

•58 

103. 

47 

"39- 

42 

.05668 

5    02 

ii 

ii.  5 

4     58     45 

108 

.29 

108 

19 

1245-36 

•09529 

4    36 

II-  5 

12 

4    46 

19 

"3 

.00 

112 

90 

1356. 

00 

3 

.13226 

4     14 

12 

TURNOUTS  WITH 

STRAIGHT 

POINT 

-RAILS  AND  STRAIGHT 

FROG-RAILS;  GAUGE  4'  8£".   See  §  265. 

Frog 
Number 
<*). 

Switch 
PointAngle 
(a). 

Length  of 
Switch 
Point 
(Z^V). 

Length  of 
Straight 
Frog-rail 

Lead  (L) 
(Eq.  90). 

Chord 
(ST) 

(Eq.  88). 

Radius  of 
Lead- 
rails 

Logr. 

Degree  of 
Curve  (d). 

Frog 
Number 

4 

3°  40' 

7.5 

1.50 

32.20 

23.09 

I25-2I 

2.09764 

47°    05' 

4 

4-5 

3     40 

7-5 

1.69 

34.29 

25.03 

159-25 

.  2O2O8 

36     36 

4-5 

5 

2     45 

IO.O 

1.87 

41.85 

29.88 

I97-65 

.29589 

29      22 

5 

5-5 

2     45 

10.  0 

2.06 

44.16 

32.03 

240.44 

.38106 

24    oo 

5-5 

6 

50 

15.0 

2.25 

56.00 

38.66 

288.09 

-45953 

19     59 

6 

6.5 

50 

15.0 

2.44 

58.84 

4L34 

340-19 

•53172 

16     54 

6.5 

7 

50 

15.0 

2.62 

61.65 

43.  c 

* 

397.65 

•  59950 

14     27 

7 

7.5 

50 

15-0 

2.81 

64-36 

46.50 

460.00 

.66276 

12      29 

7-5 

8 

50 

15-0 

3.00 

67.04 

48.99 

527-9I 

.72256 

10     52 

8 

8-5 

50 

15-0 

3.19 

69.60 

51.38 

600.94 

.77883 

9     33 

8.5 

9 

50 

15-0 

3.-  37 

72.20 

53.80 

681.16 

•83325 

8     25 

9 

9-5 

50 

15-0 

3-56 

74.70 

56.11 

767.11 

.88486 

7     28 

9-5 

10 

50 

15-0 

3.75 

77.04 

58.28 

858.14 

.93356 

6    41 

10 

10.5 

50 

15-0 

3-94 

79-51 

60.57 

959-00 

2.98182 

5     59 

10.5 

ii 

50 

15-0 

4.12 

81.82 

62.69 

1065.52 

3-02756 

5     23 

ii 

11.5 

50 

15-0 

4-31 

84.09 

64.78 

1180.16 

3.07194 

4     51 

11.5 

12 

50 

15.0 

4-50 

86.16 

66.67 

1299-93 

3-"392 

4     24 

12 

TRIGONOMETRICAL    FUNCTIONS    OF    THE    FROG   ANGLES    (F). 

Frog 

Frog 

Number  FrogAngle  (F). 

Nat.  sin  F. 

Nat.  cos  F. 

Log  sin  F. 

Log  cos  F. 

Log  cot  F. 

Log  vers  F. 

Number 

4 

14°   15'    oo" 

.24615 

.96923 

9.39126 

9 

.  98642 

10.59522 

8.4881! 

4 

4-5 

12     40    49 

.21951 

97561 

.34145 

.98927 

.64782 

.38721 

4.5 

5 

ii     25 

it 

.  19802 

, 

98020 

.  29676 

•99I3I 

.69461 

.29676 

5 

5-5 

10    23 

90 

.18033 

98360 

.25605 

.99282 

•73675 

.21467 

5.5 

6 

9     31 

38 

.16552 

. 

98621 

.21884 

•99397 

.77513 

.13966 

6 

6-5 

8     47 

51 

•15294 

.98823 

.18453 

.99486 

.81033 

.07058 

6-5 

7 

8     10 

Eti 

.14213 

98985 

.  I526§ 

-99557 

.84283 

8.00655 

7 

7-5 

7     37 

41 

•13274 

.99"5 

.12301 

.99614 

•87313 

7.94691 

7-5 

8 

7    09 

[i  • 

.12452 

99222 

.09522 

.99666 

-90138 

.89116 

8 

8-5 

6     43 

':  > 

.11724 

99310 

.06909 

.99699 

.92796 

.83864 

8.5 

9 

6      21 

35 

.11077 

993S5 

.04442 

.99732 

•78915 

9 

9-5 

6     01 

32 

•  10497 

99448 

9.02107 

•99759 

.97652 

.74232 

9-5 

10 

5     43 

-  ' 

•09975 

99501 

8.9989! 

.99783 

10.99892 

.69788 

10 

10.5 

5     27 

1 

.09502 

99548 

.9778! 

.99803 

II.O2O2I 

.65560 

10.5 

ii 

5     12 

[fl 

.09072 

99588 

•9577° 

.99826 

.  04056 

.61528 

ii 

ix.  5 

4     58     45 

.08679 

99623 

.93848 

.99836 

.05987 

•57676 

ix.  5 

12              4      46 

.08319 

99653 

8.92007 

9 

.98849 

11.07842 

7.53986 

12 

321 


TABLE    IV.— ELEMENTS    OF   TRANSITION    CURVES. 


-e- 

I 


<oo  cQ  CM  oo  o 
t^.  ON  ON  t^  co 
co  LO  vo  oo 

co  oo    «    co  LO  vo    r^  oo    o\  q 
tv.t^.odooo6oo*odcooo"    ON 


CLO  ct*x  c^~  o   cQ 

CO    vo    vo  co    *^ 

Tf       O         LO      Tj-  HH      CO 

—    vo   co     ON  ON 


0)     <U 

§  § 


ON  ct^  CT^  CON  o-i     ON   N   co 

ON  ON  ON  co  oo 
ON  ON  ON  ON  ON 
ON  ON  ON  ON  ON 


ON   ON 
ON   ON 


H-    oo     t-^  oo   cQ    <^ 

CO    "-<      M      LO    —    OO 


O     O     O 


"booooooooo 

cocoo     O     cocoO     O     coco 
"Krj    LOIOM    t-^O    O    t^M 

O      C^l      ^HI      LOCOCOCOCOLO 


t^oo    ONO 


10 


o 
7" 

b 

w 


O    o    M    LO  r-x 

*H      CO    M      Tj-     CO 

oo    n    ^t- 


8  ar& 


LO  vO   vo 


w    1-1    O 


LO  n    O    t^  LO 

^  N    co  O    - 


co    ^  i^  co  vo 

CO    CS      O      "3-     M 


LO     O       fO     •<* 


fc.       ITN     LO 

O     co    O 


"b    r^  LO  n    O 
O    O    ^-  "-o  co 


LO    CO  CO      1-1      N 

I-        O       •*      CO     M 


8M        !_/% 
LO      -I 


O     "^  CO 
l-O     CO     >-* 


-     •-,     O     O     O 


O     O     O     O 


O    r^ 
co  co 


O    O    O 


c°0 


O     O 


co   ON 
co   O 


O     - 


8M     "•> 
10      HH 


O     *•   VO 
co   O     ro 


O     HM     « 


O     O    -i    - 


S-8 


vo 


O     O    -    -i 


co     ON   M 
<->     co   O 


LO      l-l        H- 


CO     LO     M       CO 

O    O    -    ~    r 


^    co 
O     co 


8t^    LO    M      O      t^s 
O     ^    LO   co    co 


O    00    00      H- 


8M    LO  tv,    o 
H     «     co    CO 


ON   <-•    LO    rj 
•-     -<t    O     co 


O    t-*» 
co   O 


322 


TABLE    IV.— ELEMENTS   OF   TRANSITION    CURVES. 


ON   M    m   M     ON   Q    f-* 

£•  ^  iS    M   !i  &  Q 

^ .   — .   — .   ON   ON  co    t^   ^r    O    ' 
t-Ao\**O*N'i'ON^'ON'^f 

M      rtt^ONM       Tt     C^     ON     M 

(M    CvO  CM    <CO  Cf^  OO      *•*    <ON    ON 

\O   NO     M     co  VO    m    M     m 
H)  rOcOOO-'t^ONCO1^- 

oo"    O\  O\   O*    O*    O'     6    •+    •-«* 

8  O     M    tN.  vO     ON   ON  vo     ^^    in    ON 

"""        i  <O    ^  co  <co  co   ^h  co  <« 

x  i  O    CO    CO     m 

2 

T*-ir\vovo   r^r^r^oococo 

^-          <ON  <vp  xo  coo 

I  O^SN-S^O^O^SN^N 

3  O\O\ONONONONONONO\ON 

C«    CM      M    CO      O       CO  CON    CO  CCO  C( 

co     ONONVONO     ^co     COM 
ON  vO    t*x    ON   m    *i    in   *?f    O 

3          tx  oo  oo  oo  oo   oo 

§  ...     O\  c\O  cO    CCO  OO     ci-r>   f-^  oo    ceo 

*•     ONONONONCNONCOCO     I^x 
O    CNONONONONONONONON 

M 

•e- 

<co   «     M   vo    rh   m  coo  CT* 

H 


Ir 

I 


O     O    ~« 


txoo    ONO 


ce 


CO 


vbw^co    r>«<-i    O    corivo    m 
m  t*    •*$•   Q    n    cococon    >-i 


COCO     t^xt^NO     "^Tj-cOM      N- 


8  :r  ^  j?  g  -  a 

*    coo    «- 


"^O 
i-i     O 


"—     O 
w     O 


coo 


m    O     1-r*  O 

TtO     -  co 

coO     **  tN. 
O^« 


*n   O 

i-     O 


3 


"LTIQ 
—     O 


O     m   O    \n   O 
O     ^-co~     5 


O    "^   O    ""*   O     co 
corfO     •-     COT!- 


m   in   \n 


O     m   Q    in   O     m  oo     1-1 
co^tO    "-    co-^-irti-' 

MOO    OVOI^COTJ-I-I 


O     m    O     m   O    m   O     co 
O     ^co>-i     O-^-COH- 

Lncot^vo    Ooo    M    •-. 
'-'fr>"">MOcoM»- 

OOO      i-'MMCO'^- 


m   Q 

«^ 


in   O    covoco 


M     coo    ^-l^oo 


323 


TABLE    IV.— ELEMENTS    OF    TRANSITION    CURVES. 


fc 

Q   vo    t^.  vo    l\    ON   O     N   vo    rj- 
O     ON  t^   1-1   vo    to   ON   N    r-^    oo 

O     ON   ON    ON   C^    Tf  OO     ON   co    O 

x           >_     _     _     >_     r^     r^ 

1- 

A 
43 

1 
§ 

! 

13 

.52 

c 
6 

B 

s 

% 

i 

c" 
<u 

1 

s 

'5. 

a 

u 

c 

1 

1 

c 
9 

o 

s 
s 

.2 

0 

H 

*tr»    d     O     O 

o»    ^t-   O    n 

*co   O   oo    to 
10   to    CO    1-1 

fv»  VO      to    rf 

01 
01 

8 

O 

CO 

ON 

8 

vo 

0 

O 
CO 

01 

to 

8 

8 
8 

0 

Oi 

CO 

V 

CO 

IO 

ON   to    O     Q 
CO    O     co    O 

2    3£    S 
i-"     ON  oo   vo 

0 
CO 

01 
01 

8 

to 
01 

8 
8 

0 

8 

O 

CO 
M 

8 

1 

(•^f  (O    OO    (OO  (04      ON  Q^x    t^    t>x    CO 

r-x  r^  oo   vo   vo    t^.   O    cooo    ** 
t^.  vo    co  Tf  t^.   to  oj    •^•t^vo 
COCOOO     •-     t>x    ON  CO     T$-  CO     •- 
O    t^t-i    to  t^   ON  1-1    co^vo 

ON  ON  d    O    O     O    "-1    "-1    '-'    "i 

QO 

CO 

0 

0 
0 

vO     ^"    O 
co   O    co 

ON  oo    t^ 

8 

CO 

01 
CO 

8 
8 

8 

8 

o 

8 

to 

0 
CO 

CO 

6 

O      'T     OJ      t^     ON     O       O      ON    to     *- 

M    ioioc4    ON  ON   M    O    t^»   co 

O     O    "-1    *O  to   ON   to   04     O    "-• 
*i     M     CO   ^ 

- 

o"  co 

'S  2 

°ON  OQ 

0 

ft 

8 
S, 

-t- 

01 
CO 

8 

vo 

8 

8 

0 

8 
8 

01 

* 

8 

VO 

VO 

•e- 

i 

vo  co    tNi   ON  (o^    ON  03   (ON  O   (co 

VO    CO    OO     CO    CJ    OO    OO    VO     to    O 

co    cococo    coo)    rococo    to 

. 

"b 

co 

"M 

VO 

0 

8 

VO 

cT 

8 
cT 

01 

8 

8 
8 

0 

8 

ft 

CO 
CO 

8 

O 

VO 

ft 

to 

tOVO      t^    IX    J^CO    00    CO    CO      ON 

i 
i 

(OO  to  <O   (^   lx  <vo  (O   (O   d-1    co 
ON  oo    *vf   co   c^   vo    ON   c^   vO    ON 

ON   ON   ON  ON   ON   ON  oo    ^^  vo    ^^ 
ONONONONONONONONONON 

10 

"8 

e 

?0 

8 
8 

CO 

01 

01 

8 

to 

8 
8 

0 

8 

CO 

8 

VO 

-3- 

ro 

to 

VO 

to 

ON 
10 

CO 

-e- 

s 

*x 

rf    M      O    (ON    O    (CO  (tv.  OO      Tt  (O 

oo    ONCO    M    t^vovo    ONCO    •rh 
Or^COOtoOcoONnrJ- 

, 

v 

0 

CO 

8 

to 

CO 

VO 

8 
8 

8 
8 

0 

8 

CO 

8 

0 

ft 

M 

to 
to 

to  OO 

t^»    ON 

t^oooooo    ONONONONONON 

CO 

"8 
b 

0 

ft 

8 

0 

8 
8 

0 

8 

8 

o" 

01 

8 

01 
CO 

& 

01 
VO 

CN 

oo 

0 

•e- 
\    i 

ON   ON   ON   ON   ON  oo    t^   >o   04   oo 

" 

o 

o 

8 

c°0 
0 

8 
8 

0 

8 

0 

0 
CO 

CO 

8 
S- 

01 

O 
CO 

01 
VO 

CO 

oo    ^1- 
vo  vo 

VO 

ON 

04 

00 

8 

O 

-e- 

s 

at 

(t^    M   (co  (I-H   no  (04    ON   O    t^  (t^x 

- 

'8 

°o 

8 
8 

0 

8 

O 
co 

0 

cT 

8 

ft 

ON  VO 

ON   r^ 
co  to 

£ 

CO 

0 

01 
01 

CO 

0 

M 
1 

'OOOQOOOQOO 
cocoo    O    COCOQ    O    coco 

O     i-*     cototv«O    "^"OO    H    t^ 

. 

8 
"8 

0 

8 

0 

0 
CO 

CO 

0 

8 

0 

vo 

8 
-t 

01 

CO 
01 

CO 

CO 

VO 

CN 
to 

to 

oi 

ft 

•* 

VO 

VO 

co 

vO 
CO 

ON 

a 

S 

-"  —  2 

bj 

"t 

•z 

) 
rt 

O1 

H 

• 

CO 

« 

- 

- 

fc- 

« 

Ci 

0 

rH 

324 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


N. 

0     123 

4 

5    6 

7 

8 

9 

P.P. 

100 

,  101 
IO2 
I03 
104 
I°5 

106 
107 
108 
109 
110 
in 

112 

"3 
114 

H5 
116 

117 
118 
119 
120 

121 
122 

"3 

124 

I25 
126 

127 
128 
I29 

130 

131 
132 

!33 
134 
!35 
136 

137 
138 
i39 
140 
141 
142 
U3 
144 

145 
146 

147 
148 
149 
150 

oo  ooo 

043 

087 

130 

173 

2lg 

260 

303 

346 

389 

.  i 

.2 

-3 

•4 
•5 
.C 

•7 

.8 

•V 

.2 

•3 

•4 
•5 
.C 

1 

•9 

43 

,!! 

17.4 
21.7 
26.1 

3°-4 
34.8 
39.1 

46 

84:f 

12.  1 

16.2 

20.2 
24-3 

28.§ 

Zt 

31 

3-7 
7-5 

II.  2 

15-0 
18.7 
22.5 

26.2 
30.0 
33-7 

34 

3-4 
6.9 
io.| 

13.8 
17.2 
20.7 

24.  i 
27-6 
31-0 

3? 

i:! 

9-4 

12.6 

15-7 
18.9 

22.6 
25.2 
28.§ 

43 

S:l 
12.9 

17.2 

21-5 

25.8 

30.1 

34-4 
38.7 

40 

4.0 
8.0 

12.0 

l6.0 
20.0 
24.0 

28.0 
32.0 
3O.O 

37 

3-7 
7-4 
n.  i 

3:5 

22.2 

25-9 
29.6 

33-3 

34 

1.1 

10.2 

13-6 
17.0 
20.4 

23.8 
27.2 
30.6 

31 

3-1 

6.2 

9-3 
12.4 

21.7 
24.8 
27.9 

42 

,s 

16.8 

21.  O 
25-2 

29.4 

33-6 
37-8 

39 

3-9 
7-8 
11.7 

15-6 
^9-5 
»3, 

H:J 

35-i 

36 

3-6 

ioJ 

»4-4 
18.0 

21.6 

25.2 
28.8 
32.4 

33 

1:1 

9-9 

13-2 
16.5 
19.8 

23.1 
26.4 
29.7 

30 

3-o 
6.0 
9.0 

12.0 
15.0 

18.0 

21.0 

Z4.o 
17.0 

41 

C 

12.1 

16.4 
2O.  < 
24.6 

28.7 

32.8 
36.» 

3» 

3-8 
7.6 
11.4 

15.  a 

19.0 

32.8 

26.6 
30-4 

34-* 

35 

3-5 
7.0 
10.5 

14.0 

17-5 

21.0 

24-5 

28.0 

31.5 

32 

3-2 

6.4 

9.6 

12.8 

16.0 
19.2 

22.4 
25-6 
28.8 

29 

2.9 

i-1 

n.  6 
U-5 
r7-4 

20.3 
23.2 
26.1 

432 
860 

oi  283 

703 

02  119 

530 

938 
03342 
742 

475 
902 
326 

745 
166 

57i 

979 

382 
782 

5i8 

945 
368 

787 

201 

612 
*oi§ 
422 
822 

56i 
987 
410 

82§ 
243 
653 
*o6o 

463 
862 

604 
*030 
452 
870 
284 
694 

*!00 

503 
901 

646 

*072 

494 
911 
325 
735 
*i4i 

543 
941 

689 
*H4 
536 

953 
366 

77S 
*i8i 

583 
981 

*732 
*i57 

578 

994 

407 

8ig 

*22I 
623 
*020 

*775 
*i99 

619 

*o36 
448 
857 

*262 

663 

*o6o 

817 

*24I 

66  1 

*07? 

489 
898 

*302 

*7°3 

*IOO 

04  139 

i78 

218 

257 

297 

336 

375 

415 

454 

493 

532 
922 
05  308 
690 
06  070 
446 

8ij 

07  188 

554 

57i 
966 

346 

72§ 
107 

483 

855 
225 

59i 

616 

999 
384 
76g 
M5 

526 

893 
261 

627 

649 
*Q38 
423 
804 
183 
558 

93° 
298 
664 

68s 

*076 
461 

842 

220 

595 
967 

335 

706 

727 
*ii5 
499 
886 
258 
632 

*oo4 
372 
737 

*768 
*i54 

538 

9J8 
296 
670 
*04o 
4°8 
773 

80| 
*I92 
576 

956 
333 

707 

*o77 

445 
809 

844 

*23I 

614 

994 
37i 
744 
*ii4 
481 
845 

883 

*26§ 

655 

*032 

408 
78i 

*i5f 
5i8 
882 

918 

954 

990 

*02g 

*o63 

*°98 

*i34 

*i76 

*20g 

*242 

08  273 
636 
990 
09342 
691 
10  037 

383 
721 
n  059 

3U 
67£ 

*026 

377 
725 
071 

414 

755 
092 

35o 
707 
*o6i 
412 
760 
106 

448 
789 

I2g 

386 
742 
*o% 
447 
795 
146 

483 
822 
1  60 

422 

*77? 
131 

482 
830 
174 
5i7 
856 
193 

457 
8ij 
*i66 

5*7 
864 
209 

55i 
896 
227 

493 
849 

*202 

SS2 
899 

243 
585 
924 
266 

529 
884 

*237 
58g 
933 
277 
619 
958 
294 

564 
920 
*272 
621 
968 
312 

653 
99I 

327 

600 

955 
3°7 
656 

*002 
346 
687 
*02| 

361 

.1 

2 

•3 
•  4 

:l 

•9 

394 

427 

461 

494 

528 

56i 

594 

627 

661 

694 

.1 

.  2 

•3 

-4 

:i 

.<| 

i 

2 

3 
4 

I 

7 

8 
9 

727 
12  057 
385 
716 
13  033 
354 
672 
988 
14  301 

766 
096 
418 

743 
065 
386 

703 
*oi9 

332 

793 
123 

456 

775 
097 

4i7 

735 
*°5l 
364 

825 
156 
483 
807 
130 

449 

767 

*082 

395 

859 
189 

5i5 
840 
162 

481 

798 
*iij 

426 

892 

221 

548 
872 
194 
513 
830 

*i45 
457 

925 

254 
586 

904 
226 

545 
862 

*i76 

488 

958 
287 

613 

937 
258 
577 
893 

*20? 

5J9 

991 

320 
645 
969 
290 
6o§ 

925 
*239 

550 

*024 

355 

678 

*OOI 

322 

646 
956 

*270 

582 

613 

644 

675 

706 

736 

767 

798 

829 

866  891 

922 
15  229 
533 
836 
16  137 

435 

73? 
17  026 

3i§ 

952 
259 
564 
865 
165 
465 
761 

°55 
348 

983 
290 

594 
896 
J96 
494 
791 
085 
377 

*oi4 
3'3 

624 

926 
226 
524 
820 
114 
406 

*045 
35i 
655 
956 
256 
554 

849 
143 

435 

*o75 
381 
685 

987 
28g 
584 

879 
172 
464 

*iog 
412 

7i5 
*oi7 
316 
613 

9°8 

202 

493 

*i37 

442 

745 
*047 
346 
643 

938 
231 
522 

*i67~ 

473 
776 

*077 
376 
672 

967 
266 
55i 

*i98 

503 
806 

*I07 

405 
702 

997 
289 
580 

609 

638 

667, 

696 

725 

753 

782 

8ii 

840 

869 

N. 

0 

1 

2 

3    4 

5 

6 

7 

8 

9 

P.P. 

325 


TABLE    V.— LOGARITHMS    OF    NUMBERS. 


N. 

O 

1    2 

3 

4 

5 

6 

7 

8 

9 

P.  P.   ! 

150 

154 

ip 

157 
158 
159 
160 
161 
162 
163 
164 

165 
166 

167 
168 
169 
170 
171 
172 
173 
174 

176 

177 
178 
179 
180 

181 
182 
183 

184 

186 
187 
188 
189 
190 
191 
192 

194 

196 
197 
198 
199 
200 

17  609 

638 

667 

696 

725 

753 

782 

811 

840 

869 

.2 

•3 

-4 

•  5 
.6 

•  7 
.8 

•9 

. 

, 
. 

. 
.1 

.2 

•3 

•4 

•  5 
.6 

•9 

§ 
. 

i 

2 

29 

1:1 

8.7 

ii.  6 

17.4 
20.3 

23.2 

26.1 

2 

[     2 

z   5 
5    7 

i   J3 
5   15 

7   18 

3   21 

)    23 

2$ 

2.5 

7-6 

10.2 
12-7 

15-3 

17-8 
20.4 
22-9 

2 

I     2 

2     4 

3    7 

*   9 
5   ir 
5   14 

7   16 
3   18 

?   21 

22 

2.2 

2 

2 

11 
14 

16 
19 

21 

6 

6 
9 

6 
2 
9 

5 

2 

i 

2 

g 
i 

: 

1C 

•>: 

i 

i- 

2C 
22 

3 

3 
7 

s 
I 

7 

4 
8 
1 

2 

i 

3 

.8' 
.6 
•4 

.2 

.0 

.8 

.6 

•4 
.2 

K 

I 
I 

I 
• 

2 

5 

•5 

.0 

•5 

•5 
.0 

•5 

).O 

•5 

i 
i 

i 
i 

2 

2 

.  2 

-4 

27 

2.7 

10.8 
I3-5 
16.2 

18.9 

21.6 

24-3 

26 

2.6 

1:1 

5.4 

r.s 

3.2 

>.8 

3-4 

24 

7.2 

12.0 
14.4 

16.8 
19.  a 

21.6 

23 

;:i 

1.5 

5.i 
8.4 
3.7 

21 

2.1 

4-3  ! 

897 
18  184 
469 

752 
19  °33 
312 

590 
865 

20  I3§ 

926 
213 
497 
786 
06  1 
340 
617 

893 
167 

955 
241 
526 

803 
089 
368 

645 
926 
194 

984 
270 

554 

836 
117 

396 

673 

948 

221 

*OI2 

298 

582 

864 

145 
423 
706 
975 
249 

327 
611 

893 
173 

451 
J28 

276 

III 
639 
921 

201 

479 

*03Q 
303 

384 
667 

949 
229 

5°7 
783 

*o57 
330 

*I27 
412 
695 

977 
256 
534 
816 

357 

446 
724 

284 
562 
838 

*II2 
385 

412 

439 

4^6 

493 

520 

547 

574 

6oi 

623 

655 

682 

21  219 

484 

748 

22  Oil 
271 
531 

709 
978 
245 
511 
774 
037 
297 
557 
814 

736 

*oo5 

272 

537 
80  1 
063 

323 

582 
840 

J63 

298 

564 
827 
089 

349 
603 

865 

790 

*058 
325 

59° 
85| 

375 
634 
891 

*8'? 

352 
6ig 
880 
141 
401 
660 
917 

844 

*II2 

378 

643 
906 
167 

427 
686 
942 

*871 

405 
669 
932 
193 

453 
711 
968 

898 

695 
958 
219 

479 
737 
994 

924 

458 
722 
984 
245 
505 
^763 

23  °45 
299 

553 
804 

24055 
304 

55J 

797 
25  042 

285 

070 

096    121 

i47 

172 

198 

223 

249 

274 

325 
578 
829 

080 

328 
576 
822 
065 
3°9 

350 
603 

855 

I05 
353 
606 

846 
091 

334 

375 
623 
880 

129 
378 
625 
871 

358 

401 
653 
905 
154 
403 
650 

895 
382 

426 
679 

93° 
179 

427 
674 

920 
164 

704 
955 

204 

452 
699 

944 
1  88 

435 

477 

729 

980 

229 
477 
723 
96§ 

212 

455 

502 

*754 
005 

254 
502 
748 

993 
237 
479 

527 

^779 

279 
526 
773 
*oi7 
261 
5°3 

52? 

551 

575 

599 

623 

647 

672 

696 

720 

744 

768 
26  007 

245 
482 
717 

27  184 
416 
646 

792 
031 
269 

505 
746 

974 

207 

439 
669 

816 

055 
292 

529 
764 

998 
236 
462 
692 

840 

078 
316 

552 
787 

*O2I 

254 
485 
715 

863 

102 
340 
576 

811 

277 
508 

738 

887 
125 

363 

599 
834 
*o68 

300 
761 

911 

38? 
623 
858 
^091 

323 

554 
784 

935 

411 

646 
881 

*ii4 

346 

577 
805 

959 
197 

434 
670 
904 

369 
829 

983 

221 

458 

693 
928 

*i6i 

392 
623 

875 

898 

921 

944 

9^6 

989 

*OI2 

*035 

*o58 

*o86 

28  103 
330 
555 
780 
29  003 
225 

446 
665 

885 

126 
352 

578 
802 
025 
248 

68§ 
907 

149 

375 
606 

825 

048 

270 

490 
716 

929 

171 

398 
623 

847 
070 
292 

732 
956 

194 
426 
645 
869 
092 

534 
754 
972 

217 

443 
668 

892 
114 
336 

556 
776 

994 

239 
465 
696 

914 

137 

358 

578 
798 
*oi6 

262 
488 

936 
J59 

386 

606 
820 
*o38 

285 

73S 

959 
181 
402 
622 
841 

3°7 
533 
758 
981 
203 
424 
644 
863 
*o8i 

4 

7 
8 
9 

II  .2 
13-5 

15-7 

18.0 

20.2 

8.8 

II.  0 

3:J 

19.8 

8.6 

10.7 
12.9 

15.0 
17.2 

30  103 

124 

146 

168 

190 

211  |  233 

254 

276 

298 

N. 

0 

1 

2 

3 

4 

5  |  6 

7 

8 

9 

P.P. 

326 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


Px. 

012 

3 

4 

5    6 

789 

P.  P. 

200 

30  103  !  124 

146 

1  68 

190 

211 

233 

254 

276 

298 

22      OT 

2OI 

3*9  i  34i 

363 

384 

406 

427 

449 

476 

492 

513 

J        O   O 

^  & 

2y 

202 

535  556 

578 

599 

621 

642 

664 

685 

707 

728 

.2 

4-4 

.  1 
4-2 

203 

749  77i 

792 

813 

835 

856 

878 

899 

926 

941 

•  3 

6.6 

6.3| 

204 

963  !  984 

*oo5 

*027 

*o48 

*o6§ 

*09o 

*I12 

*i33 

*i54 

8.8 

8, 

205 

3i  175   196 

217 

239 

260 

281 

302 

32J 

344 

36! 

.5 

II.  0 

•4 
10.5 

206 

386  4o8 

429 

45° 

47  J 

492 

513 

534 

555 

576 

.6 

13-2 

12.6  i 

207 

597  618 

639 

660 

681 

702 

722 

743 

764 

785 

208 

805  827 

848 

869 

890 

916 

931 

952 

973 

994 

•  7 
.8 

J5-4 
17.6 

14.7 

16.8 

209 

32  014  035 

056 

077 

097 

IT8 

139 

1  60 

1  86 

201 

•9 

19.8 

18.9 

210 

222    242 

263 

284 

3°4 

325 

346 

366 

387 

407 

«A    «*,», 

211 

428 

449 

469 

490 

516 

531 

551 

572 

592 

6l3 

*vw 

*-\j 

212 

633 

654 

674 

695 

715 

736 

756 

776 

*797 

817 

.  i 

.2 

2  .O 
4.  1 

2  .O 
4.O 

2I3 

838 

858 

878 

899 

919 

940 

960 

986 

*02I 

•3 

4-J.  .  J. 

6.1 

^  .  \j 

6.0 

214 

33  041 

06  1 

082 

IO2 

122 

142 

163 

183 

203 

223 

215 

244 

264 

284 

304 

324 

344 

365 

385 

405 

425 

•4 

e 

8.2 
JO.  2 

8f. 
.0 

IO.O 

216 

445 

465 

485 

505 

525 

546 

566 

586 

606 

626 

•  j 

.6 

12.3 

12.0 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806  825 

218 

845 

865 

885 

905 

925 

945 

965 

985 

*oo4  ;*024 

.7 

.8 

14-3 

16  A 

14.0 

16  .0 

2I9 

34044 

064 

084 

104 

123 

T43 

163 

183 

203   222 

iu  ..4 

IS.  4 

18.0 

220 

242  i  262 

281  301  321 

34  1 

366 

380  400  419 

-S.        __ 

221 

439  I  459 

478 

498 

518 

537 

557  i  576 

596  6l5 

A9 

A9 

222 

635  655 

674 

694 

713 

733 

752  772 

791 

811 

.  I 

.  2 

1.9 

3Q 

1.9 

i  8 

223 

836 

850 

869 

889 

9o§ 

928 

947 

966 

986 

*°°5 

•3 

•S 

5-8 

j  •  ° 
5-7 

224 

35  025 

044 

063 

083 

102 

121 

141 

166 

179 

199 

225 

2I§ 

237 

257 

276 

295 

314 

334 

353 

372 

391 

•4 

r 

7-8 
9£ 

7.6 

9r 

1226 

411 

43° 

449 

468 

487 

5°7 

526 

545 

564 

58.3 

'.6 

•  / 
H.7 

•  b 
ii.  4  | 

227 

602 

621 

641 

660 

679 

698 

717 

736 

755 

774 

228 

793 

812 

831 

856 

869 

8^8 

907 

926  945  ;  964 

13-6 

TC  f\ 

13.3  i 

T  r   f\ 

229 

983 

*O02 

*02I 

*04o 

*°59 

*078 

*097  *u6  *i35  *i54 

•9 

I5.O 
17-5 

15.^1 
I7-I  1 

230 

36  i73 

191 

210 

229 

248 

267 

286  305  323  342 

_  c       _  rt 

231 

361 

380 

399 

417 

436 

455 

474 

i  492 

5" 

530 

*8 

IO  \ 

232 

549 

567 

586 

605 

623 

642 

66  1 

679 

698 

717 

.1 

2 

!•§ 

3*7 

i  .8  ; 
-i  6 

233 

735 

754 

773 

791 

810 

823 

847 

866 

884 

9°3 

.3 

•7 

5-5 

J-u 
5-4 

234 

921 

940 

958 

977 

996 

*oi4 

*o33 

*o5i 

*o7o 

*o8§ 

235 

37  107 

125 

143 

162 

1  86 

199 

217 

236 

254 

273 

.4   7.4 

7-2  : 

9(-\ 

236 

291 

3°9 

328 

346 

364 

383 

401 

420 

438 

456 

•  5 
.6 

*•« 

n.  i 

.0 

10.8 

237 

475 

493 

511 

530 

548 

5^6 

584 

603 

621 

639 

238 

657 

676 

694 

712 

730 

749 

767 

785 

803 

821 

.7 

12.9 

-  .   O 

12.6 

239 

840 

858 

876 

894 

912 

930 

948  967 

985 

*oo3 

•  9 

14.  o 
16.^ 

14.4 
16.2  i 

240 

38  021 

°39 

057 

07! 

°93 

iii 

129  147 

165 

183 

_  c.     —  _   '; 

241 

201 

219 

237 

255 

273 

291 

309 

327 

345 

363 

17 

*7 

242 

381 

399 

417 

435 

453 

471 

489 

5°7 

525 

543 

.  I 

1-7 

1.7  i 

243 

566 

578 

596 

614 

632 

650 

667 

685 

703 

721 

•  2 

•3 

3-5 

5-2 

5-1 

244 

739 

757 

774 

792 

810 

828 

^845 

863 

881 

*8" 

J245 

9T6 

934 

952 

970 

987 

*°°5 

*04o 

^058 

•4 

7.0 

Q  C 

6.8 

8- 

I  246 

39093 

in 

129 

146 

164 

181 

199 

217 

234 

252 

.6 

°.  7 
10.5 

•5 

10.2 

247 

269 

287 

305 

322 

340 

357 

375 

392 

410 

427 

248 

445 

462 

480 

497 

5*5 

532 

55° 

567 

585 

602 

•7 

12.2 

II-9 

249 

620 

63? 

655 

672 

689 

707 

724 

742 

759 

776 

.8 

.0 

I4.O 
15  .7 

13-6 
15.3 

250 

794 

811 

823 

846  863 

881 

898  915 

933 

950 

V 

.X- 

0 

1 

2 

3    4 

5 

6    7 

8 

9 

P.P. 

327 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


1  N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P, 

250 

39  794 

811 

823 

846 

863 

881 

898 

91! 

933 

95° 

251 
252 
253 
254 
255 
256 

257 
258 
259 

967 
40  140 
312 

483 
654 
824 

993 
41  162 

984 

329 
500 
671 
841 

*OIO 

179 

346 

*002 
174 

346 

688 
858 

i9§ 
363 

191 

363 
534 
705 
875 

212 

383 

*°36 
209 

380 

722 
892 
*o6i 
229 
397 

*°54 
226 

398 

569 
739 
9°8 

*07? 

246 

071 
243 
415 
586 
756 
925 

*°94 
263 

430 

266 
432 
603 

773 
942 
*ni 

279 

447 

277 
449 
620 
79° 
959 

*I2g 

296 
464 

295 
466 

637 

807 

976 
*i45 
3*3 

480 

.1 

.2 

•3 

•4 
•  5 
.6 

3-5 

5-2 

7.0 

8-7 
10.5 

17 

3-4 
5-1 

6.8 

8-5 

10.2 

260 

497 

514 

530 

547 

564 

581 

597 

614 

631- 

647 

.8 

12.2 
I4.O 

ii.  9 

13.6 

261 

;262 
263 
264 
265 
266 

267 
268 

664 
830 

995 
42  160 

324 
488 

651 

813 

683 
846 

*OI2 
177 
341 

5°4 
667 
829 

69? 
863 

*02§ 
193 

35? 

683 
846 

714 
880 

209 
373 
537 
700 
862 

736 

896 
*o6i 

226 

39° 
553 
716 
878 

747 
o  1  3 
"078 

4°6 
569 
732 
894 

764 
*92? 

259 

423 
586 

748 
910 

786 
946 
•in 

275 
439 
602 

765 

927 

797 
962 

292 

455 
613 

943 

813 
979 
*i44 
308 
472 
635 
79? 
959 

•9 
.1 

.2 

•  3 

A 

15.7 

16 

1-6 
3-3 
4.9 

6  6 

15-3 

16 

1.6 
3-2 
4.8 

6  4 

269 

975 

991 

*oo7 

023 

040 

^056 

072 

*o88 

104 

*I20 

.5 

8.2 

8.0 

270 

43  J36 

155 

163 

184 

200 

216 

233 

249 

265 

28l 

.6 

9.9 

9.6 

271 
272 
273 
274 

275 
276 

277 
278 
279 

,297 

457 
6ig 

775 
933 
44091 

248 
404 
566 

473 
632 

791 

949 
log 

263 

420 
576 

329 
489 
648 
8og 
965 

122 
279 

435 

345 
505 
664 

822 
980 
138 

295 

45  * 
607 

361 

520 
680 

838 
996 

316 
467 
622 

377 
536 
695 
854 

*OI2 
l6§ 

482 
638 

393 
552 
711 

870 

*028 

185 

342 
498 

653 

409 

56§ 

727 

886 
*°43 

2OI 

357; 

669 

425 
584 
743 
901 

*°5? 

373 

529 
685 

441 
606 

759 
917 

*°75 
232 

389 

545 
706 

•  7 
.8 

•9 
.1 

.2 

.3 

ii.  5 
13-2 
14-8 

1-5 
4-6 

II.  2 
12.8 
14.4 

IS 

3-0 
4.5 

280 

716 

731 

747 

762 

778 

793 

809 

824 

839 

855 

A 

6  2 

6  o 

281 
282 
283 
284 
285 
286 
287 
288 
289 

876 
45  025 
178 

484 

636 

788 

939 
46  090 

886 
040 
194 

347 
499 
652 

803 
954 
I05 

901 

209 
362 

667 

8ig 
969 

120 

917 
071 

224 

37? 
530 
682 

833 
984 

932 
o8g 
240 

393 

545 
697 

848 
999 
150 

948 

102 

255 
408 
566 
712 
864 

*oi4 
165 

963 
117 

270 

423 

576 

72? 
879 

*02§ 
1  80 

978 
132 
286 

438 
743 

*891 

i95 

994 
148 
301 

454 
606 

758 
909 
*°59 

210 

i6| 

469 
621 

773 
924 

*°75 
225 

•  5 
.6 

•  7 
•9 

7-7 
9-3 

10.  § 

12.4 
13-9 

14 

1.4 

7-5 
9.0 

10.5 

12.  0  i 
13.5 

I  4. 

290 

240 

255 

269 

284 

299 

314 

329 

344 

359 

374 

.2 

2.8  ; 

291 
292 
293 
294 

295 
296 

297 
298 
299 

53§ 
687 

834 
982 
47  129 

275 

421 

567 

404 

553 
701 

849 
997 
144 

290 
436 
581 

419 
568 

864 
*OII 

596 

434 

583 

879 

*02g 
173 

319 
465 

610 

449 
59? 
746 

*894 
^041 

188 

334 
480 
625 

464 
612 
.761 

9°8 

*°55 
202 

348 
494 
639 

479 
627 

775 

*O7<5 
217 

363 
5°9 
654 

493 
642 
790 

*938 
232 
378 
523 
66§ 

5°8 
657 

805 

*IOO 

246 
392 
538 
683 

523 
672 
820 
967 
*H4 
261 

407 

552 
697 

•3 

•4 

.5 

•  7 

.8 

•9 

4-3 

5.8 

7-2 

8-7 

IO.I 

ii.  6 
13.6 

4.2  : 

5.6 
7.0 

8.4 

9.8 

II.  2 
12.6 

300 

712 

726 

741 

755 

770 

784 

799 

813 

828 

842 

N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P 

328 


TABLE    V.— LOGARITHMS    OF    NUMBERS. 


X. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P 

J300 

47  712 

726 

74i 

75S 

770 

784 

799 

813 

828 

842 

301 
302 
3°3 

304 

305 
306 

307 
308 

309 

856 
48  006 

144 

287 

430 
572 

714 

855 

996 

871 
°i5 
J58 
3°i 

444 
58g 
728 
869 

*OIO 

885 
029 
173 
316 

458 
606 

742 
883 

*O24 

900 
044 

187 

330 
472 
614 

756 
897 

*o38 

914 

053 

201 

344 
487 
629 

770 
*9li 

*052 

928 
072 
216 

358 
5°i 
643 
784 
925 
*o66 

943 

087 
230 

373 

5i| 
657 

798 
939 
*o8o 

957 

101 

244 
387 

529 
671 

812 

^ 
094 

972 
IJ5 
259 

401 

543 
685 

827 
967 
*io8 

98g 
130 

273 

415 

558 
699 

841 
982 

*I22 

.1 

.2 

•3 

14 

1.4 
2.9 

4-3 

c  ft 

M 
1.4 

2.8 

4.2 

e  fi 

310 

49  J36 

J5o 

164 

178 

192 

206 

220 

234 

248 

262 

•5 

5-° 
7.2 

5-°  1 
7.0 

3" 
312 

3'3 
3'4 

3i5 
316 

3i7 
3i8 
3i9 

276 

4i? 

554 

693 
831 
96§ 
50  106 
242 
379 

290 
429 

568 
707 

845 
982 

119 

256 

392 

304 

443 
582 

726 

858 
996 

133 

270 

4°6 

3i8 
457 
596 

734 

872 

*OIO 

147 
283 
420 

332 
471 

610 

748 
886 

*023 

166 
297 
433 

346 
485 
624 

762 
900 
*037 
174 
3H 
447 

359 
499 
637 
776 

*9'3 
*o5i 

1  88 

324 
466 

373 
5'3 
65! 
789 
927 
*o65 

2OI 
33? 

474 

387 
526 
665 

803 
941 
*o?8 

215 
352 
488 

4OI 
540 
679 

817 

*955 
*092 

229 

365 

501 

.6 

•  7 
.8 

•9 

8-7 

10.1 
n.  6 
13.6 

8-4 
9.8 

II.  2 
12.6 

1320 

5i5 

528 

542 

555 

569 

583 

596 

610 

623 

637 

i3 

13 

321 
322 
323 
324 
325 
326 

327 

!328 

!329 

650 

78$ 

920 

51  °54 

i8§ 
322 

455 
58? 
7i§ 

664 
799 
933 
068 

2OI 

335 
468 
600 
733 

677 
812 
947 
081 

2I5 
348 
481 
614 
746 

691 
826 
966 
094 

228 
36? 

494 
627 

759 

704 

839 
974 
1  08 
242 

375 
508 
646 

772 

718 

853 
987 

121 

255 
38§ 
S2! 
653 

785 

73i 
86g 

*OOI 

J35 

263 
401 

534 
667 

798 

745 
880 
*oi4 
148 
282 
415 

547 
680 
812 

758 
893 

*027 

161 

295 

428 

56i 
693 

825 

772 
907 

*o4i 

175 
3o§ 
441 

574 
706 
838 

.1 

.2 

•3 

•4 
!6 

•  7 

.8 

•9 

1.3 

2-Z 

4.0 

5-4 

6.7 

8.1 

9.4 
10.8 

12.  1 

1-3 
2.6 

3-9 

5-2 
6-5 

7-8 

9.1 
10.4 
ii.  7 

,830 

851 

864 

877 

891 

904 

917 

930 

943 

956 

969 

331 
332 
333 
334 
335 
336 

337 
338 
339 

983 
52  114 
244 
374 
5°4 
634 

763 
891 
53020 

996 
127 
257 
387 
5'7 
647 

776 
904 

033 

*oo9 
140 

276 
406 

530 
660 

789 
917 

045 

*022 

!53 

283 

413 

543 
672 

Soi 
930 

058 

*o35 
166 

296 

42§ 
556 
685 

814 

943 
071 

*048 
179 

309 
439 
569 
698 
827 

956 
084 

*o6I 
192 
322 

452 
582 
711 

840 

968 
097 

*074 
205 
335 
465 

595 
724 

853 
981 
109 

*o87 

2I§ 

348 

478 
608 

737 
866 
994 

122 

*IOO 

231 
361 
491 
621 

750 
879 
*oo7 
135 

.1 

.2 

•  3 

12 

1.2 

2-5 
3-7 

12 

1.2 
2.4 

3.6! 

A  R 

340 

148 

1  66 

173 

1  86 

199 

211 

224 

237 

250 

262 

•4 

•  5 

•° 

6.2 

4.8  ! 
6.0 

34i 
342 
343 
344 
345 
346 

347 
348 
349 

275 
402 

529 
656 
782 
907 

54033 
158 
282 

288 

4i5 
542 

66§ 

794 
920 

045 
170 

295 

301 
428 
554 
681 
807 
932 
058 
183 
3°7 

3i3 
446 

567 

693 
819 

945 

076 

193 

320 

326 
453 
580 

7og 
832 

958 
083 
208 
332 

339 
466 

592 
719 

845 
976 

095 

220 

344 

352 

478 
605 

73? 
857 
983 
108 
232 
357 

364 
491 
618 

744 
870 

995 
126 
245 
369 

377 
5°4 
636 

756 
882 
*oo8 

133 

257 
382 

39° 
5*6 
643 
769 
895 

*020 

14! 

270 

394 

.6 

!s 

•9 

7.5 

8.7 

IO.O 
II.  2 

7.2 

8.4  1 
9.6 
10.8 

350 

407 

419 

43  i 

444 

456 

469 

481 

493 

506 

5r8 

X. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P 

329 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

350 

54  407 

419 

43i 

444 

456 

469 

481 

493 

506 

5i8 

T3 

35  1 

530 

543 

555 

568 

580 

592 

605 

617 

629 

642 

m  i 

1  .  2 

352 

654 

665 

679 

691 

703 

716 

728 

740 

753 

765 

.2 

2.5 

353 

777 

790 

802 

814 

825 

839 

851 

863 

876 

888 

•3 

x 

3-7 

354 

906 

912 

925 

937 

949 

961 

974 

986 

998 

*oio 

5r\ 

355 

55  023 

035 

047 

059 

071 

084 

096 

I0§ 

120 

133 

•4 

.5 

.U 

6.2 

356 

M5 

i57 

169 

181 

194 

206 

218 

236 

242 

254 

.6 

7-5 

357 

267 

279 

291 

3°3 

3i5 

327 

340 

352 

364 

376 

8C 

353 

38§ 

400 

412 

424 

437 

449 

461 

473 

485 

497 

•7 

.8 

•  7 

IO.O 

359 

5°9 

521 

533 

545 

558 

570 

582 

594 

606 

618 

•9 

II.  2 

860 

636 

642 

654 

665 

678 

696 

702 

7i4 

726 

738 

T  *•» 

361 

756 

762 

775 

787 

799 

811 

823 

835 

847 

859 

m.m 
I/> 

362 

871 

883 

895 

907 

919 

93i 

943 

955 

966 

978 

.2 

•  ** 

2.4 

363 

996 

*O02 

*oi4 

*02g 

°38 

*°5<3 

*o63 

*o74 

*o86 

*o98 

•3 

3.6 

364 

56  no 

122 

134 

146 

158 

170 

181 

T93 

205 

217 

A  9, 

365 

229 

24I 

253 

265. 

277 

283 

306 

312 

324 

336 

•4 

.5 

4.0 
6.0 

366 

348 

360 

372 

383 

395 

407 

419 

43i 

443 

455 

.6 

7-2 

367 

465 

478. 

496 

502 

5H 

525 

537 

549 

56i 

573 

8. 

368 

585 

596 

603 

620 

632 

643 

655 

667 

679 

691 

•  7 
.8 

•4 
o  6 

369 

702 

714 

726 

738 

749 

761 

773 

785 

796 

8o§ 

•9 

v-u 

10.8 

370 

820 

832 

843 

855 

867 

879 

896 

902 

914 

925 

M« 

37i 

937 

949 

961 

972 

984 

996 

*oo7 

*oi9 

*Q3i 

*042 

X  X 

1C 

372 

57  054 

066 

077 

089 

101 

112 

124 

136 

147 

*59 

.  i 

.2 

.  I 
2  3 

373 

171 

182 

194 

2O6 

217 

229 

246 

252 

264 

275 

•3 

•  J 

3-4 

374 

287 

299 

310 

322 

333 

345 

357 

3^8 

380 

39i 

.    £ 

375 

403 

414 

426 

438 

449 

461 

472 

484 

49! 

507 

•4 

t  r 

4.6 

59 

376 

5i9 

530 

542 

553 

565 

576 

588 

599 

611 

622 

.6 

•  / 
6.9 

377 

634 

645 

657 

66§ 

680 

691 

703 

714 

726 

737 

378 

749 

766 

772 

783 

795 

8og 

818 

829 

841 

852 

•7 
.8 

8.6 

9n 

379 

864 

875 

887 

898 

909 

921 

932 

944 

955 

967 

•9 

.4 
10.3 

380 

978 

99° 

*OOI 

*OI2 

*024 

*035 

*047 

*°58 

*o6§ 

*o8i 

38i 

58  092 

104 

"5 

125 

138 

149 

161 

173 

183 

195 

JL1 

382 

20§ 

217 

229 

246 

252 

263 

274 

286 

297 

308 

.  i 

.  2 

I  .  I 
2  .  2 

383 

320 

33i 

342 

354 

365 

376 

388 

399 

416 

422 

•3 

3-3 

384 

433 

444 

455 

467 

478 

489 

5°i 

:*2 

523 

535 

385 

546 

557 

568 

580 

59i 

602 

613 

625 

636 

647 

•4 

4.4 

386 

658 

670 

681 

692 

703 

7i5 

726 

737 

748 

760 

'.6 

5-5 
6.6 

387 

771 

782 

793 

804 

816 

827 

838 

849 

861 

872 

388 

883 

894 

9°5 

9i6 

928 

939 

95° 

961 

972 

984 

•  7 

7-7 

8Q 

389 

995 

*oo6 

*oi7 

023 

*03§ 

*°5o 

^062 

*o73 

*o84 

*095 

_ 

-9 

.  O 

Q  Q 

390 

59  106 

117 

I2g 

140 

T5i 

162 

173 

184 

!95 

20g 

_  ** 

39i 

217 

229 

240 

251 

262 

273 

284 

295 

306 

3J7 

IO 

392 

32§ 

339 

351 

362 

373 

384 

395 

406 

4i7 

428 

.1 

1.6    7 

2T 

393 

439 

455 

46l 

472 

483 

494 

505 

5*6 

527 

538 

•  2 

.3 

*  JL 

3.1 

394 

549 

566 

571 

582 

593 

604 

615 

628 

637 

648 

395 

659 

676 

681 

692 

703 

714 

725 

736 

747 

758 

•4 

4.2 

c  5 

396 

769 

786 

791 

802 

813 

824 

835 

846 

857 

868 

•  5 
.6 

5'2 
6-3 

397 

879 

890 

901 

912 

923 

933 

944 

955 

966 

977 

398 

98§ 

999 

*OIO 

*O2I 

*032 

*043 

*°53 

*o64 

*o75 

*o8g 

•  7 

7-3 
8. 

399 

60  097 

108 

119 

130 

141 

151 

162 

173 

184 

195 

.8 
•  Q 

•  4 

Q.4 

400 

206 

217 

227 

238 

249 

266 

271 

282 

293 

303 

§Tr     ^  • 

N. 

O 

1 

2 

3 

4 

5    6 

7 

8 

9 

P.P. 

330 


TABLE   V.— LOGARITHMS    OF    NUMBERS, 


X. 

O 

1 

2 

3 

4 

5 

e 

7 

8 

9 

P 

.  P. 

400 

60  206 

217 

227 

238 

249 

266 

27! 

282 

293 

303 

4oi 

402 

4°3 

404 

405 
406 

314 

422 

533 
638 

745 
852 

325 
433 
54i 
649 
756 
863 

336 
444 

659 
767 

874 

347 
455 
563 
676 

777 
884 

357 
466 

573 
681 

788 
895 

3^8 
476 
584 
692 

799 
906 

379 

487 

595 
702 
810 

390 
498 
606 

826 
927 

401 

5°9 
6ig 

724 
831 
938 

412 

627 

735 
842 

949 

.1 

.2 

-3 
•4 

II 
i.i 

2.2 

3-3 

4.4 

!407 
|4o8 
1409 

959 
61  066 
172 

970 

076 
183 

981 
087 
193 

99  1 
098 
204 

*002 
I0g 

215 

'013 
119 

225 

*02J 

130 
236 

146 
246 

257 

'^bb 
161 
268 

'.6 

5-5 

6.6 

410 

278 

289 

299 

310 

320 

331 

342 

352 

363 

373 

.8 

7-7 
8.8 

411 
412 

414 

415 
416 

417 
418 
419 

384 
489 
595 
700 
805 
9°9 
62  013 
117 

221 

394 
5°o 
605 

710 

920 
024 
128 
232 

405 

5" 

616 

721 
825 
93° 
034 
^38 
242 

416 

62g 

836 
946 

045 
149 

252 

426 
532 
637 
742 
846 
95  i 
055 
T59 
263 

437 
542 
647 

752 
857 
961 

065 
169 
273 

447 
553 
658 

763 
867 
972 
076 
1  80 
283 

458 

563 
66§ 

773 
878 
982 

o8g 
196 
294 

4^8 
574 
679 

784 
88g 
993 
097 

200 
3°4 

479 
584 
689 

794 
*8" 
107 

211 
314 

«9 
.1 

.2 

•3 

•4 

.5 

9-9 

id 

i.o 

2.1 

3.1 

4-2 

5.2 

420 

325 

335 

345 

356 

366 

376 

387 

397 

407 

4l8 

.6 

6-3 

421 

42*2 
423 
424 
J425 
426 

427 
429 

428 

531 
634 

736 
839 
941 

63  043 

144 

245 

438 
54? 
644 

747 
849 

95  1 
053 

256 

449 

552 
654 
757 
859 
961 

063 
164 
266 

459 
562 
665 

767 
869 
971 

073 

175 
276 

469 
572 
675 
777 
879 
981 

083 

185 
286 

480 
582 
685 

788 
890 
992 

°93 
195 
296 

49° 
593 
695 

798 
900 

*002 
104 
205 

506 
603 
706 
808 
916 

*OI2 
114 

215 

5*0 

716 

8i§ 

926 

*022 
124 
22§ 

326 

521 
624 

82g 

93  1 

032 

134 

235 
336 

.'s 

•9 
.1 

.2 

•3 

7-3 

8.4 

9-4 

10 

I.O 
2.0 

3-0 

430 

347 

357 

367 

377 

387 

397 

407 

4J7 

427 

437 

432 
433 
434 
435 
436 
437 
438 
439 

447 
548 
649 

749 
849 

948 
64  048 
147 
246 

458 
558 
659 
759 
859 
958 
058 

256 

468 

568 
669 

769 
869 
968 
068 
167 
266 

478 

578 
679 

779 
879 
978 
078 
177 
-276 

488 

689 

789 

889 

088 
187 
286 

498 

598 
699 

799 
899 
998 
098 
197 
296 

508 

6og 
709 

809 

909 
*oog 

107 

207 
306 

6i§ 
719 

819 
919 

*OIg 

117 
217 

315 

528 

729 
829 
928 

*02g 

127 

325 

538 
639 
739 
839 
938 
*o38 

137 

236 
335 

•4 
-5 
.6 

!a 

•9 

4.0 
5-0 
6.0 

7-0 
8.0 
9.0 

9 

440 

345 

355 

365 

375 

384 

394 

404 

414 

424 

434 

.1 

.2 

0.9 

441 
442 
443 
444 
445 
446 

447 
449 

444 
542 
646 

738 
836 

933 
65  031 
128 

224 

453 
552 
650 

748 
846 

943 
046 
137 
234 

463 
562 
660 

758 
855 
953 
050 

244 

473 

670 
767 
865 
962 
060 
157 
253 

483 
58? 
679 

777 
875 
972 
069 

i6g 
263 

493 

689 
787 
885 
982 
079 
176 
273 

503 

601 
699 

797 
894 
992 
089 
186 
282 

5" 
611 

709 

8og 
9°4 

*OOI 

098 

195 
292 

522 
621 

8ig 
914 
*oi  i 
log 
205 
302 

532 
636 

728 
826 
923 

*02I 

118 

215 
311 

-3 
•4 
'.6 

Is  • 

•9 

3.8 
4-7 

5-7 

6.6 
7.6 

8.5 

450 

321 

331 

340 

35° 

360 

369 

379 

389 

398 

408 

X. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P, 

P. 

331 


TABLE    V.— LOGARITHMS    OF    NUMBERS. 


!  ]ST. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

.P. 

450 

65  321 

33i 

340 

35° 

360 

369 

379 

389 

398 

408 

45  1 
452 
453 
454 
455 
456 

457 
458 
459 

4i7 
5*4 
610 

7°3 
80  1 

896 
991 
66  085 
181 

427 
523 
619 

7i5 

8i5 
906 

*OOI 

096 
190 

437 
533 
629 

724 
820 

9i5 

*OIO 

i°S 

20O 

446 
542 

638 

734 
830 

925 

*020 

H5 
20§ 

456 
552 
648 

744 
839 
934 

*O29 

124 
219 

466 
562 
657 
753 
849 
944 

*°39 
134 

22§ 

475 
57i 
667 

763 
858 
953 

*04§ 

143 

238 

485 
58i 
677 

772 
868 

963 

*o58 
153 

247 

494 

59° 
68g 

782 

877 
972 
*o67 
162 

257 

5°4 
600 
696 
791 
887 
982 

*o77 
172 
26g 

.1 

.2 

•3 

•4 

•  5 
.6 

10 

I.O 
2.0 
3-0 

4.0 
5-0 

6.0 

460 

276 

285 

294 

304 

313 

323 

332 

342 

351 

366 

•  7 
.8 

7.0 
8.0 

461 
462 
463 
464 
465 
466 

467 
468 
469 

370 
464 
558 
652 
745 
838 

93l 
67  024 

ii? 

379 
473 
567 
661 

754 
848 

941 
034 
I26 

389 

483 

577 
670 
764 
857 

956 
043 
136 

398 
492 
586 
680 

773 
865 

959 

052 

145 

408 
502 

595 
689 
782 
876 
969 
061 
154 

417 

511 

605 

698 
792 

885 

978 

07! 

163 

426 
526 
614 
708 
80  1 
894 

98? 
080 

173 

436 
530 
623 
717 
816 
904 

996 

089 
182 

445 
539 
633 

72g 

820 
9J3 
*oo6 
099 
191 

455 
548 
642 

736 
829 
922 

*OI5 
108 

200 

>  -9 
.1 

.2 

•  3 
•4 

9.0 

8 

0.9 

*-3 

2-8 

*•* 

470 

210 

219 

22g 

237 

246 

256 

265 

274 

283 

293 

.6 

5-7 

47i 

472 
473 
474 
475 
476 

477 
478 

1479 

302 

394 

486 

578 
669 
766 

852 

943 
68033 

3Ji 

403 
495 
587 
678 
770 

861 

952 

042 

320 

412 

5°4 

596 
687 

779 
870 
961 
051 

329 

422 

5i3 
605 

697 

788 

879 
970 

060 

339 
43i 
523 
614 
706 
797 
888 

979 

070 

348 

440 
532 

623 

715 

Sog 

897 
988 
079 

357 
449 
54i 

633 

724 
8i5 
9°6 
997 
088 

366 
458 
556 
642 

733 

824 

9i5 
*oog 
097 

376 
467 

559 

651 

742 
833 
924 

*oi5 
106 

385 

477 
568 
660 

75? 

842 

*93? 

^024 

"5 

•  7 
.8 

•9 
.1 

.2 
•  3 

6.g 
7.6 

8.5 

0.9 
1.8 
2.7 

480 

124 

i33 

142 

151 

160 

169 

178 

187 

196 

205 

481 
482 
483 
484 
485 
480 

487 
488 
489 

214 
304 
394 

484 

574 
663 

753 
842 

93i 

223 
313 

403 
493 
583 
672 

762 

851 
940 

232 
322 
412 
502 

592 
681 

776 
860 
948 

241 

33i 

421 

5ii 
601 

696 

779 

86§ 

957 

250 
340 
435 
526 
610 
699 

78§ 
877 

96§ 

259 
349 
439 

529 
619 

7°8 

797' 
885 
975 

263 

358 
448 

538 
628 
717 

805 
895 
984 

277 
367 
457 
54? 
637 
726 

815 
904 
993 

285 
376 
46g 

556 
646 

-735 
824 

9i3 

*002 

295 
385 
475 
565 
654 
744 

833 
922 

*oio 

•4 
•  5 
.6 

'.8 
•9 

3-6 
4-5 
5-4 

6.3 
7.2 

8.1 

490 

69  019 

028 

037 

046 

°55 

064 

073 

081 

096 

099 

.1 

2 

0.8 
j  i 

491 
492 
493 
494 

495 
496 

497 
498 
499 

108 

196 
284 

372 

466 
548 

635 
723 

810 

117 

205 
293 
38? 
469 

557 
644 

73? 
819 

126 
214 

302 

390 
478 
56g 

653 

740 
82? 

'34 
223 

3n 

399 

487 
574 
662 

749 
836 

143 
232 

320 
408 
495 
583 
676 
758 
845 

155 
246 

32§ 

4^6 

5°4 
592 

679 
768 

853 

161 

249 
337 

425 

5i3 
606 

688 

775 
862 

170 

258 
346 

434 
522 
609 

697 

784 
871 

179 
267 

355 

443 
53o 
618 

7o§ 
792 
879 

187 
276 
364 
45? 
539 
627 

714 
80  1 
88§ 

•3 
•  4 
[6 

•  7 

.8 

•9 

2-5 

3-4 

4.2 

5-1 

5-9 
6.8 

7-6 

500 

897 

90! 

914 

923 

93i 

946 

949 

958 

96g 

975 

N. 

0 

1 

2 

3 

4 

5 

O 

7 

8 

9 

P. 

T> 

332 


TABLE    V.— LOGARITHMS    OF    NUMBERS. 


!  X. 

0 

1 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

1500 

69  897 

905 

914 

923 

93i 

946 

949 

958 

966 

975 

501 

502 

5°3 
5°4 

5°5 
506 

507 
508 

5°9 

984 

70  076 

157 
243 
329 

415 

501 
586 
672 

992 
079 

i6| 

337 
423 

5°9 

595 
686 

*OOI 

087 
174 

266 

346 
432 

603 
689 

*OIO 

°96 
182 

269 

355 
441 

526 
612 
697 

*oi§ 

191 
277 
36j 
449 

535 
626 
706 

113 

200 
286 
372 

458 

543 
629 
714 

*o36 

122 
20§ 

294 
386 
466 
552 
637 
723 

217 

303 
389 
475 
566 
646 

"053 

226 
312 
398 
483 

569 
654 
740 

*o6i 
148 
234 
326 
406 
492 
578 
663 
748 

.1 

.2 

•3 

•4 

•  5 
.6 

9 

0.9 

1.8 
2.7 

3-6 
4-5 
5-4 

6  3 

510 

757 

765 

774 

782 

791 

799 

808 

8ig 

825 

833 

.8 

7-2 

511 

512 

5*5 
5*7 

iST9 

842 
927 
71  oil 

096 
1  86 
265 

349 
433 

856 
935 

020 

105 
189 

273 

357 
441 

525  i 

859 
944 

02§ 

113 
197 
282 

366 

449 

533 

867 
952 
037 

121 

206 
290 

374 

458 
542 

876 
961 

045 
130 
214 
298 
382 
466 
550 

884 
969 
054 

138 
223 

3°7 

391 

475 
558 

893 
978 
062 

147 
23I 

399 
483 
567 

901 

986 
071 

239 
324 

408 
491 

575 

910 

995 
079 

164 

248 
332 
416 

500 
583 

9J8 

*°°3 
088 

172 

256 
340 
424 

5°8 
592 

•9 
.1 

.2 

-3 

•4 

.5 

8.1 

8 

2.5 

3-4 
4.2 

520 

600 

6  03 

617 

625 

633 

642 

656 

659 

667 

675 

.6 

5-1 

521 

522 
523 
524 

525 
526 

527 
528 
529 

684 
767 
850 

933 
72  016 

°98 
181 
263 

345 

692 

775 
858 
941 
024 
107 
189 
271 
354 

706 

783 
867 

949 

032 

197 
280 
362 

709 
792 
875 
958 
046 
123 
206 
288 
370 

717 
806 
883 

966 
049 

214 
296 

378 

725 
803 
891 

974 
057 
140 

222 
3°4 

734 
817 
900 

983 
065 
148 
236 
312 
395 

742 
825 
9°8 
991 
074 

238 
321 

403 

833 

999 
082 
164 
247 

329 
411 

758 
842 

925 

096 
i73 

255 
337 
419 

'.8 
•9 

.1 

.2 
•  3 

5-9 
6.8 

7-6 

8 

0.8 
1.6 
2.4 

530 

427 

436 

444 

452 

466 

46§ 

476 

485 

493 

501 

532 
533 

5°9 

5*2 

599 
681 

526 
607 
689 

534 
697 

624 
705 

55° 
632 

558 
640 
721 

566 
648 
729 

575 
656 
738 

583 
664 
746 

•4 

•5 

•i 

3-2 
4.0 
4.8 

e  fi 

534 
535 
536 

537 
538 
539 

754 
835 
9X6 
997 
73  078 
159 

762 
843 
924 

*°°5 
o8g 
167 

770 
932 

094 
'75 

778 
859 
941 

*02I 

io2 

868 
949 

*°3° 
no 
191 

795 

876 

957 

*o3S 

"8 

199 

803 
884 

965 

207 

811 
892 
973 

215 

819 
906 
981 

*062 

22J 

827 

9°8 
989 

^070 
151 
231 

.8 
•9 

6.4 

7.2 

f 

1540 

239 

247 

255 

263 

271 

2/9 

287 

295 

303 

3V 

.1 

0.7 

542 
543 
544 
545 
546 

547 
548 
549 

3^9 

400 
480 
560 

639 
719 

798 
878 

957 

328 
408 
488 
568 
647 
727 
8og 
886 

965 

336 
416 
496 

576 
655 
735 
814 
894 
973 

344 
424 
5°4 

584 
663 

743 
822 
902 
981 

352 
432 

592 
671 

751 
836 
909 
989 

360 
440 
520 
600 
679 
759 

838 
917 

997 

368 
448 
528 
608 
687 
767 

846 

376 
456 
536 

695 
775 
854 
933 

*OI2 

384 
464 

544 
623 
703 
783 
862 
941 

*O2O 

392 
472 
552 
631 
711 
791 
870 
949 

*02§ 

•  3 
•4 
'.6 

is 

•9 

2.2 

SI 

4-5 

5-2 
6.0 

6-7 

550 

74  036 

044 

052 

060 

068 

075 

083 

091 

099 

107 

X. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

.P.     | 

333 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8  I 

9 

P. 

P- 

550 

74  035 

044 

052 

060 

068 

075 

083, 

091 

099 

107 

551 

552 
553 
554 
555 
556 

557 
558 
559 

**5 

194 

272 

351 
429 

5°7 

585 
663 
741 

123 

202 
286 

359 

437 
51! 

593 
671 

749 

131 

209 
288 

366 
445 
523 
601 
679 
756 

139 

217 
296 

374 
453 
53i 
609 
687 
764 

146 

225 

3°4 
382 
466 

538 
6ig 
694 

772 

*54 
233 
312 

39° 
4^8 
546 
624 

702 
780 

162 
241 
3*9 

398 

476 

554 
632 
710 

788 

170 
249 
327 
406 

484 
562 

640 
718 
795 

178 
257 
335 

4i3 
492 

57o 
648 

725 
803 

186 
264 
343 
421 
499 
577 
655 
733 
811 

.1 

.2 

•3 

8 

0.8 
1.6 
2.4 

560 

819 

825 

834 

842 

850 

857 

865 

873 

881 

88s 

•4 

5 

3-2 
4.0 

561 

562 
563 
564 
565 
566 

567 
568 

569 

896 
973 
75  051 

128 
205 
281 

358 
435 
511 

904 
981 
°58 
135 

212 
28§ 
366 
442 

5T9 

912 
989 
065 

M3 
220 
297 

373 
45° 
526 

919 

997 
074 

J5T 
228 

3°4 
38i 
458 
534 

927 

*oo4 
081 

158 
235 
312 

389 
465 
54i 

935 

*OI2 

089 

166 
243 

320 

396 
473 

549 

942 

*O2O 
097 

174 
25J 

327 

404 

486 

557 

95o 

*027 

105 
182 

258 
335 

412 
483 
564 

958 
*°35 

112 
l8§ 
26g 

343 
419 
496 

572 

966 
*Q43 

120 
I97 
274 
35° 
427 

503 
580 

.6 

'.8 
•9 

4.8 

5-6 
6.4 
7.2 

570 

587 

595 

602 

616 

618 

625 

633 

641 

648 

656 

A 

57i 

572 
573 
574 
575 
i576 

577 
1578 
579 

663 
739 
8i| 
891 
967 
76  042 
117 

i93 

268 

671 

747 
823 

899 
974 

050 

i25 

200 

275 

679 

755 
836 

9°6 
982 

057 
132 
208 
283 

6ig 
762 
838 
914 
989 
065 
140 

215 
296 

694 
770 
846 
921 

997 

072 

M7 

223 
298 

701 
777 
853 

929 

^004 
080 

J55 
236 

305 

709 

785 
86  1 

936 

*OI2 
087 
162 
238 
313 

7i7 

792 

86§ 

944 

*OI§ 

°95 
170 

245 
326 

724 
806 
876 

95i 

*027 
102 
I78 

253 

328 

732 

808 
88j 

959 
*034 
no 

185 

266 
335 

.1 

.2 

•  3 
•4 
'.6 

•  7 
.8 

•9 

7 
0.7 

'•Jj 

2.2 

3-o 

3-7 
4-5 

5-2 

6.0 
6.7 

580 

343 

35° 

358 

365 

372 

380 

387 

395 

402 

410 

5«i 
1582 

583 
584 
585 
586 

587 
588 

589 

417 
492 
567 
641 

7i5 
790 

864 

937 
77  oil 

425 
500 

574 

648 
723 
797 
871 

945 
019 

432 
5°7 
582 

656 

730 
804 

878 
952 
025 

440 
514 
589 
663 
738 
812 

886 
960 
033 

447 

522 

596 
671 

745 
819 

893 
967 
041 

455 

529 
604 

678 

752 
827 

901 
974 

°48 

462 

537 
611 

686 
760 

^34 
908 
982 
°55 

47° 
544 
619 

693 
767 
841 

9T5 
989 
063 

477 
552 
626 

706 

775 
849 

923 
997 

076 

485 
559 
634 
708 
782 
856 

93o 

*oo4 
078 

.1 

.2 

•  3 

7 
0.7 
1.4 

2.1 

590 

085 

092 

IOO 

107 

114 

122 

129 

!36 

144 

i5i 

•4 
.  5 

2  .  0 

3  •  5 

591 
592 
593 
594 
595 
596 

597 
598 
599 

158 
232 

3°5 
378 

45i 
524 

597 
670 

742 

166 
239 

3*3 

386 

459 
532 
604 
677 

750 

173 

247 
320 

393 
466 
539 
612 

684 

757 

181 

254 
327 
400 
473 
546 
619 
692 
764 

188 
261 

335 
408 
481 

554 

625 
699 
771 

J95 
269 

342 

415 

488 

561 

634 
706 
779 

203 
276 

349 

422 

495 
568 
641 

7i3 
786 

210 
28J 

356 
43° 
5°3 

575 

648 
721 

793 

217 
291 
364 
437 
510 
583 

655 
728 
800 

225 
298 
37i 
444 
5*7 
59° 
663 

735 
808 

.6 

'.8 
•9 

4.2 

4-9 
t.6 

6-3 

600 

815 

822 

829 

837 

844 

8gj 

858 

866 

873 

880 

I  N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

.P. 

334 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


X. 

o 

1 

2 

3 

4 

o 

6 

7 

8 

9 

P. 

P. 

600 

77  815 

822 

829 

837 

844 

85i 

858 

866 

873 

880 

601 
602 
603 
604 
605 
606 
607 
608 
609 

887 

959 

78031 

103 

'?! 
247 

3i9 

39° 
461 

894 
967 
039 
in 
182 
254 
326 

397 
469 

902 

974 
046 

118 
190 
26! 

333 

404 
476 

909 
981 
053 

125 
197 
269 

340 
412 

483 

9*6 
9*S 
066 

132 
204 
276 

347 
419 

49° 

923 
995 
067 

i39 

211 

283 

354 

426 

497 

*931 
003 

075 
147 

2I§ 

290 

362 

433 

5°4 

938 

*OIO 

082 

154 
226 
297 

369 

446 

51* 

945 

*oi7 
089 

161 
233 
3°4 
376 
447 
5'8 

952 

*024 

°96 
i6§ 
240 

3ii 

383 
454 
526 

.1 

.2 

-3 

? 

0.7 
i-5 

2.2 

610 

533 

540 

547 

554 

561 

568 

575 

583 

59° 

597 

•4 

.5 

3-9 
3-7 

611 
!6l2 

613 

614 

615 
|6i6 

617 
618 

619 

604 

675 
746 

817 
887 
958 
79  023 
099 
169 

611 
682 
753 
824 
894 
965 

035 
106 
176 

613 
689 
760 

831 

901 
972 
042 
H3 
183 

625 

696 
767 

838 
9°8 
979 
049 
1  20 
190 

632 

7oj 
774 

845 
91! 
986 

°56 
127 

197 

639 

716 
781 

852 
923 
993 
063 

J34 

204 

646 
717 

78§ 

859 
93° 
*ooo 

076 
141 

211 

654 
725 

795 
86g 

937 
*oo? 

078 
148 
218 

661 

732 
802 

873 
944 
*oi4 

085 

J55 
225 

668 

739 
810 

886 
95i 

*02I 
092 
162 
232 

.6 

Is 

•9 

4-5 

5-2 
6.0 
6.7 

620 

239 

246 

253 

260 

267 

274 

28l 

288 

295 

302 

621 

|622 

3°9 
379 

316 
386 

323 
393 

330 
400 

337 
407 

344 
414 

351 
421 

358 
428 

365 
435 

372 
442 

.1 

.2 

0.7 
1.4 

;623 
1624 
625 
1626 
627 
628 
629 

449 

5'8 

588 

657 

727 
796 
865 

456 
525 
595 
664 

733 

803 
872 

462 

532 
602 
671 

740 
810 
879 

469 

539 
609 
678 

747 
8ig 
886 

476 

546 
616 
685 

754 
823 
892 

483 

553 
622 
692 
761 
830 
899 

490 
560 
629 
699 

76§ 
837 

9°6 

497 
567 
636 
706 

775 
844 
9iJ 

5°4 
574 
643 
7i3 
782 

85! 
926 

5-M 

581 
656 
720 

789 
858 

927 

•3 
-4 
'.6 

'.S 
•9 

2.1 

2.8 

3.5 

4-2 

4.9 

5-6 
6-3 

630 

934 

941 

948 

954 

961 

968 

975 

982 

989 

996 

631 

632 

633 
634 

635 
636 

637 
638 

639 

80  003 
071 
140 
209 
277 
345 
414 
482 
550 

010 

0/8 
147 
216 
284 
352 
421 
489 
557 

oi<3 
085 

154 

222 
291 

359 
427 
495 
563 

023 
092 
161 
229 
298 
366 

434 
502 

570 

030 
099 
168 

236 
304 
373 
441 

5°9 

577 

037 
1  06 

J74 
243 
3lJ 
380 

448 
5i6 
584 

044 

"3 

181 

250 
3T8 
3S6 
455 
523 
59i 

051 
120 

183 

257 
325 
393 
461 

529 
597 

058 
I26 
*95 
263 

332 

406 

463 

536 

604 

065 
J33 

202 

270 

339 
407 

475 
543 
611 

.1 

.2 

•3 

8 

0.5 

T'3 
1.9 

610 

618 

625 

631 

638 

645 

652 

658 

665 

672 

679 

•4 

.5 

2.6 

3.2 

641 
642 

643 
644 

645 
646 

647 
648 
649 

686 

753 
821 

883 
956 
81  023 

090 

J57 
224 

692 
766 
828 

895 
962 
030 
097 
164 
231 

699 
767 
834 
902 
969 

036 
104 
171 

238 

706 

774 
841 

909 
976 
043 
no 
177 
244 

7i3 

786 
848 

9T5 

983 

050 

117 
184 
251 

719 
787 
855 
922 
989 
057 
124 
191 
258 

726 
794 
86: 

929 

996 
063 

136 
197 
264 

733 
801 

863 

936 
*oo3 
076 

137 
204 
271 

740 
807 
875 
942 

*010 

077 
144 

211 

278 

746 
814 
882 

949 

*0lg 

083 

I5i 

218 
284 

.6 

•  7 

.8 

•9 

3-9 

4.5 
5-2 

5-8 

650 

291 

298 

3°4 

3n 

3i8 

324 

33i 

338 

345 

35i 

X. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

,  P. 

335 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

.  P.     i 

650 

81  291 

298 

304 

31* 

3i8 

324 

33i 

338 

345 

35i 

'651 
1652 

:653 
1654 

655 
656 

657 

;658 

659 

358 

425 
491 

558 
624 
690 

756 

822 

883 

365 
43i 
498 

564 
631 
697 

763 
829 

895 

37i 
438 
5°4 
57i 
637 
703 
770 
836 
901 

378 
444 
5lJ 

577 
644 
710 

776 

842 

9°8 

385 
45i 
5i8 

584 
656 
717 

783 
849 
9i5 

39i 

458 
S2! 
591 
657 
723 
789 

855 
921 

398 
464 

53i 

597 
664 

730 
796 
862 
928 

405 
471 

538 
604 
676 
736 
803 
869 
934 

411 

478 
544 
611 

677 
743 
809 

875 
941 

418 
484 
55i 
6if 

684 

75° 
816 
882 
948 

.1 

.2 

•  3 

7 
0.7 
1.4 

2.1 

660 

954 

961 

967 

974 

986 

987 

994 

*ooo 

*oo7 

*oi3 

•4 

2.8 

661 
662 
663 
664 
665 
666 
667 
668 
;669 

82  020 
086 
I5! 
217 
282 

347 
412 

477 
542 

025 

092 

158 

223 

283 

354 
419 
484 
549 

°33 
099 

164 

230 

295 
366 

425 
496 

555 

040 

I05 
171 

236 
302 

367 
432 
497 
562 

046 

112 
177 

243 

308 
373 
438 
503 
56§ 

053 

"8 
184 

249 
3*5 
380 

445 
5io 

575 

°59 
1*5 

196 

256 
321 

386 
45  i 
5J6 
581 

066 

131 
197 

262 
328 
393 
458 

523 
588 

072 

138 
203 

269 
334 
399 
464 

529 
594 

079 
J45 

210 

275 
341 
406 

471 
53<5 
60  1 

.6 

!s 

•9 

•  0 

4.2 

4.9 
5-6 
6.3 

670 

607 

614 

626 

627 

633 

640 

646 

653 

659 

666 

2 

671 
672 
673 
674 

675 
'676 

677 
1678 
679 

672 

737 
80  1 

866 
93o 
994 

83  059 
123 
187 

678 
743 
808 

872 
*937 

*00  1 

065 
129 
193 

685 

75° 
814 

879 
943 

*oo7 

071 
136 

200 

691 

756 
821 

885 

949 
*oi4 

078 
142 
206 

698 

763 

827 

892 
956 

*O2O 
084 

148 

212 

704 
769 
834 
898 
962 

*O27 

091 

155 

219 

711 

775 
840 

904 
969 
*033 
097 

i6i 

225 

717 

782 

846 
911 

*975: 

*°39 
103 
168 
231 

724 
78§ 
853 
917 
982 
*046 
no 

174 
238 

730 
795 
859 

924 

988 

*052 

IJ6 
186 

244 

.1 

.2 
•3 

•  4 

.  c 

.6 

•  7 

.8 

•9 

6 

o.g 

J:l 

2.6 

3.2 

3-9 

4-5 
5-2 

5-8 

680 

25  i 

257 

263 

270 

276 

283 

289 

295 

302 

3°8 

681 
682 
683 
684 

685 
686 
687 
688 
689 

3H 
378 
442 

5°5 
569 
632 

695 
759 

822 

321 

385 
448 
512 
575 
638 
702 

765 
828 

327 
39i 
455 

5i8 
58i 
645 

708 
771 

834 

334 
397 
461 

524 
S88 

651 

7H 

778 
841 

340 
404 

467 
531 

594 
657 
721 
784 
847 

346 
416 

474 

537 
606 
664 

727 
796 

853 

353 
4i6 
480 

543 
607 

676 

733 
796 
859 

359 
423 
48g 

55° 
613 
676 

740 
803 
866 

365 

429 

493 

556 
619 
683 
746 
809 
872 

372 
435 
499 
562 
626 
689 

752 
815 
878 

.1 

.2 

•3 

6 

0.6 

1.2 

1.8 

690 

885 

891 

897 

904 

910 

9^6 

922 

929 

935 

941 

•  5 

•4 

3-o 

691 
692 

693 

694 

695 
696 

697 
698 
699 

948 
84  016 

073 
136 

198 
261 

323 

385 
447 

954 
017 
079 
142 
204 
267 

329 
392 

454 

966 
023 
086 

148 

211 

273 

335 
398 
460 

966 
029 
092 

J54 
217 
279 

342 

404 

465 

973 
035 
°98 
161 
223 
286 

348 
416 
472 

979 

042 
104 
167 
229 
292 

354 
4^6 
479 

985 
048 
ii  i 

173 
236 

298 
366 

423 

485 

992 

054 
117 

179 
242 

3°4 
367 
429 

49  1 

998 
061 
123 

186 

248 
3u 
373 
435 
497 

*oo4 
067 
129 
192 
254 
317 

379 
441 

503 

.6 

•  7 

.8 

•9 

3-6 

4.2 
4.8 

5-4 

700 

5io 

5i6 

522 

528 

534 

54i 

547 

553 

559 

565 

i  N' 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P, 

P, 

336 


TABLE    V.— LOGARITHMS    OF    NUMBERS. 


X. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

•  P- 

700 

84  510 

5i6 

522 

528 

534 

54i 

547 

553 

559 

565 

701 
702 
703 
704 

705 
706 

707 
708 
709 

572 

633 

695 

757 
819 
886 

942 

85  003 
064 

578 
640 
701 

763 

825 
885 

948 
009 
076 

584 
646 
708 

76§ 
831 
893 
954 

OI5 

077 

590 
653 
7M 
776 

837 
899 

966 

021 
083 

596 
658 
720 

782 
843 
9°5 

966 

028 
089 

603 
664 
726 
788 

849 
911 

972 
034 
095 

609 
671 

732 

794 
856 
917 

979 

040 

ioi 

6i5 
677 
739 
806 
862 

923 

985 
046 

107 

621 
683 
745 
805 
868 
929 
991 
052 
"3 

627 
689 
75i 
8i3 
874 
936 

997 
°5  8 
119 

.1 

.2 

•3 

§ 

0.6 

1:1 

710 

126 

132 

138 

144 

150 

'56 

162 

i6§ 

i74 

181 

•4 

C 

2.6 

3  2 

711 
712 
7i3 
7H 

7i5 
716 

717 
718 
719 

187 
248 
3°9 
37o 
435 
491 

552 
612 

673 

193 
254 
3i5 
376 
436 
497 
558 
6i§ 
679 

199 
260 
321 

382 
443 
503 

564 
624 
685 

205 
26g 
327 
388 

449 
509 
570 
636 
691 

211 

272 

333 
394 

455 
51! 
576 

636 
697 

217 
278 
339 
400 
461 
521 
582 
642 
703 

223 
284 

345 

4°6 
467 

S2? 
588 

648 
709 

229 
296 

351 

412 
473 
533 
594 
655 
7i5 

236 
297 
357 
4i§ 
479 
540 
606 
66  1 
721 

242 
303 
36j 
424 
485 
546 
605 
667 
727 

.6 

•  7 
.8 

•9 

3-9 

4-5 
5-2 

5-8 

720 

733 

739 

745 

75i 

757 

763 

769 

775 

78i 

787 

721 

722 
723 
724 

725 
1726 

(727 
728 
729 

793 
853 
914 

974 

86  034 

093 

153 
213 

273 

799 
859 
920 

980 
040 
099 

'59 
219 

278 

805 
865 
926 
986 
046 
105 

165 

225 
284 

811 
872 
932 
992 

°5? 
in 

171 
231 

296 

817 
878 
938 

998 
058 
117 
177 
237 
296 

823 
884 

944 
*oo4 
063 
123 

183 
243 
302 

829 
890 
950 

*OTO 
069 
I2§ 
189 
249 

3°8 

835 
896 

956 
*oi6 
075 
I35 
*95 
255 
3i4 

841 
902 
962 

*022 
08  1 
141 
2OI 
26l 
320 

847 
908 
968 

*028 

087 

147 
207 
267 
326 

.1 

.2 

•3 
•4 
.6 

'.S 
•9 

0.6 

1.2 

1.8 

2.4 
3-0 
3-6 

4.2 
4.8 
5-4 

730 

332 

338 

344 

35o 

356 

362 

368 

374 

380 

386 

73i 
732 
733 
734 
735 
736 

737 
738 
739 

39i 
45i 
5io 
569 

62g 

688 

746 

805 
864 

397 
457 
5i6 

575 
634 
693 
752 

8ii 
876 

403 
463 
522 

58? 
646 
699 

758 
817  i 
876 

409 
469 
528 

587 
646 
705 
764 
823 
882 

4i5 
475 
534 

593 
652 
711 
776 
829 
888 

421 
481 
540 
599 
658 
717 

776 
835 
894 

427 
48g 
546 
605 
664 
723 
782 
841 
899 

433 
492 
SS2 
611 

670 
729 

788 
847 
9°5 

439 
498 
558 

6i7 
676 

735 
794 
852 
911 

445 
5°4 
563 
623 
682 
74i 
800 

858 
917 

.1 

.2 

•3 

S 

o.§ 
i.i 
i-6 

740 

923 

929 

935  ! 

941 

946 

952 

958 

964 

970 

976 

•  4 

.5 

2.2 

2  7 

74i 
742 
743 
i744 

745 
746 

,747 
748 
749 

982 
87  046 
099 

157 
215 
274 
332 

390 
448 

987 

046 

104 
163 

221 

279 

338 
396 

454 

993 
052 
no 
169 
227 
285 

343 

402 
460 

999 
058 

"6 

'75 
233 
291 

349 

407 

465 

*oo5 
064 

122 

1  86 

239 
297 

355 
4ij 
47i 

*OII 

069 
128 
i8g 
245 
3°3 

361 
419 

477 

*oi7 
075 
134 
192 
256 
309 
367 
425 
483 

*023 
081 
140 
198 

256 
3i4 
372 

43i 
489 

*02g 
087 

J45 
204 
262 
326 

378 
436 
494 

*034 
°93 
!5i 

2IO 
268 
326 

384 
442 
506 

.6 

'.& 
•9 

3.3 

3-8 
4-4 
4.9 

750 

5o6  | 

5i2 

5*7 

523 

529 

535 

54i 

546 

552 

558 

X. 

O 

1  1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

337 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


•  N. 

0 

1 

2    3  |  4 

5 

6 

7 

8 

9 

P.  P. 

750 

87^06 

512 

5i7 

S2! 

529 

535 

54i 

546 

552 

558 

75i 

564 

57o 

575 

58J 

587 

593 

598 

604 

610 

616 

752 

622 

627 

633 

639 

645 

656 

656 

662 

668 

673 

753 

679 

685 

691 

697 

702 

7°8 

7H 

720 

725 

73i 

754 

737 

743 

748 

754 

760 

766 

771 

777 

783 

789 

755 

794 

806 

806 

812 

8i7 

823 

829 

835 

846 

846 

!756 

852 

858 

863 

869 

875 

881 

885 

892 

898 

904 

6 

•757 

909 

9i5 

921 

927 

932 

938 

944 

949 

955 

961 

.1 

.  2 

0.6 

1  .2 

1758 

967 

972 

978 

984 

99° 

995 

*ooi 

*oo7 

*OI2 

*OI§ 

•  3 

i.'s 

759 

88  024 

030 

035 

041 

047 

053 

058 

064 

070 

075 

760 

081 

087 

°93 

°98 

104 

no 

JI5 

121 

127 

133 

•4 

.5 

2.4 

a  .0 

761 

138 

144 

!5° 

155 

161 

167 

172 

178 

184 

190 

.6 

J*V 

3-6 

762 

195 

201 

207 

212 

2I§ 

224 

229 

235 

241 

247 

763 

252 

258 

264 

269 

275 

281 

285 

292 

298 

303 

Q 

4.2 

A  ft 

764 

309 

315 

326 

326 

332 

337 

343 

349 

355 

366 

.  O 

•9 

4-0 
5-4 

765 

366 

372 

377 

383 

389 

394 

400 

406 

411 

417 

766 

423 

42§ 

434 

440 

445 

45i 

457 

462 

468 

474 

767 

479 

485 

49  1 

496 

502 

508 

5*3 

5T9 

525 

530 

768 

536 

542 

547 

553 

558 

564 

57o 

575 

58i 

587 

769 

592 

598 

604 

609 

6i5 

621 

625 

632 

638 

643 

770 

649 

654 

666 

666 

671 

677 

683 

683 

694 

700 

Q 

771 

705 

711 

7^6 

722 

728 

733 

739 

745 

75° 

756 

5 

772 

761 

767 

773 

778 

784 

79° 

795 

801 

805 

812 

.1 

o-5 

773 

818 

823 

829 

835 

846 

846 

85! 

857 

863 

86§ 

.  2 

•3 

i  .  i 

!-6 

774 

874 

879 

885 

891 

896 

902 

907 

9*3 

919 

924 

775 

93° 

936 

941 

947 

952 

958 

964 

969 

975 

986 

•4 

2,2 

776 

986 

992 

997 

*oo3 

*oog 

*oi4 

*OI§ 

*025 

*°3I 

*o36 

'.6 

2-7 

3-3 

777 

89  042 

047 

053 

°59 

064 

070 

075 

081 

087 

092 

!778 

098 

103 

109 

114 

120 

126 

!3i 

'37 

142 

148 

-7 

3-8 

!?79 

153 

J59 

165 

170 

I76 

181 

187 

193 

J98 

204 

.  8 

.0 

4.4 

A.  0 

780 

209 

215 

220 

226 

23I 

237 

243 

248 

254 

259 

*  "       -f  ~  s       \ 

781 

265 

270 

276 

282 

287 

293 

298 

3°4 

3°9 

3i5 

782 

326 

326 

332 

337 

343 

348 

354 

359 

3-65 

37o 

783 

376 

38i 

387 

393 

398 

404 

409 

4i5 

426 

426 

784 

43i 

437 

442 

448 

454 

459 

465 

476 

476 

481 

785 

487 

492 

498 

5°3 

5°9 

&f4 

520 

S2! 

53i 

536 

786 

542 

548 

553 

559 

564 

570 

575 

58i 

58g 

592 

5 

787 

597 

603 

6o§ 

614 

619 

625 

636 

636 

641 

647 

.  I 

0.5 

788 

652 

658 

663 

669 

674 

680 

685 

691 

696 

702 

.2 

.  -3 

i  .0 

i  •  ^ 

789 

707 

7i3 

7i§ 

724 

729 

735 

746 

746 

75i 

757 

o 

•*•  D 

790 

762 

768 

773 

779 

784 

790 

795 

80  1 

806 

812 

•4 

r 

2.0 

n  K 

791 

817 

823 

823 

834 

839 

845 

856 

856 

86  1 

867 

.'6 

*•  D 

3-o 

1792 

872 

878 

883 

889 

894 

900 

9°5 

911 

9J6 

922 

793 

927 

933 

938 

943 

949 

954 

960 

965 

971 

976 

3-5 

794 

982 

987 

993 

998 

*oo4 

*oo9 

*oi§ 

*020 

*026 

*°3I 

•9 

4.0 

A.S 

795 

9°  °36 

042 

047 

053 

°58 

064 

069 

075 

086 

086 

796 

091 

097 

IO2 

107 

JI3 

JI8 

I24 

129 

J35 

146 

797 

146 

i5i 

i56 

162 

167 

173 

178 

184 

189 

195 

798 

200 

205 

211 

2lg 

222 

227 

233 

238 

244 

249 

799 

254 

260 

26J 

271 

276 

282 

287 

292 

298 

303 

800 

3°9 

3H 

320 

325 

330 

336 

34i 

347 

352 

358 

N. 

0 

1 

2 

3  |  4 

5  |  6 

7 

8 

9 

P.P. 

338 


TABLE   V.— LOGARITHMS    OF   NUMBERS. 


n^r 

o 

1    2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

800 

80  1 
802 
803 
804 
805 
806 
807 
808 
809 

810 

811 
812 
813 
814 

815 
816 

817 
818 
819 
820 
821 
822 
823 
824 

825 
826 

827 
828 
829 
880 

831 

832 

833 
834 

835 
836 

837 
838 

839 
840 

841 
842 
843 
844 

845 
846 

847 
848 

849 
850 

90  309 

3£4 

320 

325 

330 

336 

34i 

347 

355 

358 

$ 

.1   0.5 

.2    I.I 

-3   i-6 

.4    2.2 

.5   2.7 
.6   3-3 

•  7   3-8 
.8   4-4 
.9   4.9 

5 

.1   0.5 

.2    I.O 

•  3   1-5 

.4    2.0 

•5   2.5 
.6   3.0 

•7   3-5 
.8   4.0 

•9   4-5 

363 

417 
471 
525 

579 
633 
687 
74i 
795 

368 
423 
477 
53i 
585 
639 
692 

746 

800 

374 

42§ 

482 

536 
59° 
644 
698 

752 
805 

379 
433 
488 

542 
596 
649 
703 
757 
811 

385 
^439 
493 

547 
60  1 

655 

7°9 
762 

8ig 

390 
444 
498 

552 
605 
666 

7H 

768 
821 

396 
45° 
5°4 

558 
612 
666 
719 

773 
827 

401 
455 
5°9 

563 
617 
671 

725 
778 
832 

406 
466 

515 

569 
622 

676 

73o 
784 
838 

412 
466 
520 

574 
628 
682 

736 
789 
843 

848 

854 

859 

864 

870 

875 

886 

886 

891 

896 

902 

955 
91  009 

062 
116 
169 

222 
275 
328 

907 
961 
014 
068 

121 

174 
227 
286 

333 

9!3 
966 
019 

073 
125 

179 

233 
286 

339 

918 
971 

025 

078 
J3i 
185 
238 
291 

344 

923 
977 
036 

084 

137 

196 

243 
296 
349 

929 
982 
036 
089 
142 
J95 

249 

302 

355 

934 
987 
041 
094 
147 

201 
254 

3°7 
360 

939 
993 
046 

IOO 

153 

205 

259 
312 

365 

945 
998 
052 

105 
158 

211 
264 
318 
371 

95s 
*ooj 

057 
no 
163 
217 

270 

323 
376 

381 

38g 

392 

397 

402 

408 

413 

418 

423 

429 

434 
487 

540 
592 

64! 
698 

756 
803 

855 
908 

439 

492 
545 

598 
656 

703 
756 
8og 
866 

445 
497 
55° 
603 
656 

7°8 
761 

813 
866 

45° 
503 
556 
6o§ 
661 

7U 

76g 
819 

871 

455 
5°8 
56i 
614 
665 
719 

771 
824 
876 

461 
5*3 
566 
619 
671 
724 

777 
829 
881 

466 

5'9 

57i 
624 
677 
729 
782 

834 
887 

47i 
524 
577 
629 
682 
735 
787 

839 
892 

476 
529 

582 

635 

687 

740 
792 
845 
897 

482 
534 
587 
640 
692 

745 
798 
850 
902 

9i3 

9J8 

923 

92§ 

934 

939 

944 

949 

955 

960 

92  OI2 

064 

116 

i6g 
226 
272 

324 
376 

965 
017 
069 

122 

174 
226 

277 

329 
381 

970 
023 
075 
127 
179 
231 
283 
335 
386 

976 
028 
080 
132 
184 
236 
288 
340 
39i 

981 

033 
085 

137 
189 
241 

293 

345 
397 

986 

038 
096 

142 

194 
246 

298 

350 
402 

991 

043 
096 

148 

200 
252 

303 

355 
407 

996 
049 

101 

153 

205 

257 

309 
366 
412 

*002 
054 

106 

158 

210 
262 

3M 

366 
417 

*oo7 

059 
iii 

163 

215 
267 

3i9 
37i 

423 

428 

433 

438 

443 

448 

454 

459 

464 

46§ 

474 

479 
53i 
583 

634 
685 

737 

78§ 

839 
891 

485 

536 

588 

639 
691 

742 

793 
844 
896 

49° 
54i 
593 

644 
696 

747 

798 
850 
901 

495 
546 
598 
649 
701 
752 
803 

855 
906 

506 

552 
603 

655 

706 

757 
809 
860 
911 

5°5 
557 
6o§ 
660 
711 
762 
814 
865 
9*6 

5*o 
562 

613 
665 

7^6 

768 

819 

876 
921 

515 
567 
619 

670 
721 
773 
824 
875 
926 

521 

572 
624 

675 
727 
778 
82§ 

886 
93i 

526 

577 
629 

686 
732 
783 

834 
885 

937 

942 

947 

952 

957 

962 

967 

972 

977 

982 

988 

X. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

339 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


I  N. 

0 

\ 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P.   ! 

850 

92  942 

947 

952 

957 

962 

967 

972 

977 

982 

988 

851 
852 

853 
854 

855 
856 

857 
858 

859 

993 
93  °44 
°95 
146 

196 

247 

298 
348 
399 

998 
049 

100 

151 

201 

252 

303 

354 
404 

*°°3 
054 

!05 

156 

207 

257 
308 

359 
409 

*oo§ 

059 
no 

161 

212 

262 

313 

364 
414 

013 
064 
J15 
166 
217 
267 

3'8 

369 
419 

*oi§ 
069 

120 
171 

222 

272 

323 

374 
424 

*023 

074 
125 

*76 

227 
278 

328 
379 
429 

*02§ 

079 
136 

181 
232 
283 

333 
384 
434 

*034 
084 

135 
185 

237 
288 

338 
389 
439 

*°39 
090 
146 
191 
242 
293 
343 
394 
445 

.1 

.2 
•3 

3 

o.§ 
i.i 
1-6 

860 

45° 

455 

460 

465 

470 

475 

480 

485 

490 

495 

-4 

C 

2.2 

29 

86  1 
862 
863 
864 
865 
866 
867 
868 
869 

500 

55° 
601 

651 

7oi 

752 
802 

852 
902 

5°5 
556 
606 

656 

7°6 
757 
807 

857 
907 

510 
56i 
611 

661 
71! 

762 
812 
862 
912 

5*5 
566 
616 

665 

7i6 
767 

8i7 
867 
917 

526 

57i 
621 

671 
721 
772 
822 
872 
922 

525 
576 
626 

676 
726 

777 
827 
877 
927 

530 
58i 
63i 
681 

731 

782 

832 
882 
932 

535 
586 

636 
685 
736 
787 

837 
887 

937 

540 

59i 
641 

691 
742 
792 
842 
892 
942 

545 
596 
646 
696 
747 
797 

847 
897 

947 

.6 

•  7 

.8 

•9 

3-3 

3-8 
4.4 
4-9 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

871 
872 
873 
874 

875 
876 

877 
878 
879 

94  002 
051 

101 

151 

201 
256 
300 

349 
399 

007 

°56 
log 

156 
206 

255 
305 
354 
404 

012 
O6  I 
III 

161 

210 
266 
310 

359 
409 

017 
065 
"6 
166 

215 
265 

3i5 
364 
4i3 

022 
071 
121 
171 
220 
270 
320 
369 

4i8 

025 
076 
I26 
176 

225 

275 
324 
374 
423 

031 
081 
!3i 
181 
236 
280 

329 

379 

423 

036 
085 
136 

1  86 

235 
285 

334 
384 
433 

041 
091 
141 
191 
246 
290 

339 

389 
438 

°46 
°96 
146 

196 
245 
295 
344 
394 
443 

.1 

.2 

•3 
•4 
.6 

•  7 
.8 

•9 

5 

0.5 

I.O 

1.5 

2.0 

2-5 
3-0 

3-5 
4.0 

4-5 

880 

448 

4.S3 

458 

463 

468 

473 

478 

483 

487 

492 

881 
882 
883 
]884 
885 
886 
887 
888 
889 

497 
547 
596 

645 
694 

743 
792 
841 
890 

502 

552 
601 

650 
699 

748 

797 
846 

895 

5°7 

556 
606 

655 
704 

753 
802 

851 
900 

512 
56i 
611 

660 
709 
758 
807 
856 
9°5 

5*7 
56g 
6i5 
665 
714 
763 
812 
861 
909 

522 

57i 
626 

670 
719 
768 
817 
865 
914 

527 

576 
625 

674 
724 
773 
821 
876 
919 

532 
58? 
630 
679 
728 
777 
825 
875 
924 

537 
586 

635 
684 

733 
782 

83! 
886 

929 

542 

591 
646 

689 

738 
787 

836 

885 

934 

.  i 

.2 

•  3 

4 

0.4 

si 

890 

939 

944 

949 

953 

958 

963 

96§ 

973 

978 

983 

•  4 

e 

1.8 

2  2 

891 
892 
893 
894 

895 
896 

897 
898 
899 

988 

95  °36 

085 

J34 
182 
231 
279 

327 
376 

992 
041 

090 

138 
187 

23? 
284 
332 
38i 

997 
046 

095 

143 
192 

246 
289 
337 
385 

*002 

05  T 
099 

148 
197 

245 
294 
342 
390 

*oo7 
056 
104 

153 

201 

250 

298 
347 
395 

*OI2 

061 
109 

158 

2og 
255 
303 
352 
4OO 

*oi7 
o6£ 
114 

163 

211 

260 

3o§ 
356 
405 

*O22 
076 
II9 

16? 

216 

264 
313 

361 

410 

*02g 

075 
124 

172 

221 
269 

318 

366 

414 

031 
086 
129 

177 
226 

274 
323 
37i 
419 

.6 

'.8 
•9 

2.7 

3-1 

3-6 
4.0 

900 

424 

429 

434 

438 

443 

448 

453 

458 

463 

467 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

,P. 

340 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


jr. 

O 

1 

2 

3 

4 

5    6 

7 

8 

9 

P.  P. 

900 

95  424 

429 

434 

438 

443 

448 

453 

458 

463 

467 

901 

472 

477 

482 

487 

492 

496 

501 

506 

511 

5i6 

902 

526 

525 

530 

535 

540 

544 

549 

554 

559 

564 

9°3 

569 

573 

578 

583 

5B8 

593 

597 

602 

607 

612 

904 

617 

621 

625 

631 

636 

641 

$45 

656 

655 

660 

9°5 

665 

669 

674 

679 

684 

689 

693 

698 

703 

708 

906 

7J3 

717 

722 

727 

732 

737 

741 

746 

751 

756 

907 

766 

765 

776 

775 

780 

784 

789 

794 

799 

804 

,908 

8og 

813 

818 

823 

827 

832 

837 

842 

847 

851 

909 

856 

86  1 

866 

876 

875 

886 

885 

890 

894 

899 

910 

904 

909 

9i3 

9'8 

923 

928 

933 

937 

942 

947 

911 

95  2 

956 

961 

966 

971 

975 

986 

985 

99° 

994 

5 

912 

999 

*oo4 

*oo9 

*oi4 

*OI§ 

*02J 

*02S 

*033 

*037 

*042 

.1   0.5 

.2     I  .O 

9i3 

96  047 

052 

056 

06  1 

066 

071 

075 

086 

085 

090 

•3   i-5  • 

1914 

094 

099 

104 

109 

H3 

H8 

123 

128 

132 

137 

9i5 

142 

147 

151 

156 

161 

166 

176 

175 

1  80 

185 

.4    2.O 

50  f 

916 

189 

194 

199 

204 

20§ 

213 

218 

222 

227 

232 

*•  D 

.6    3.0 

9i7 

237 

241 

246 

251 

256 

266 

265 

270 

275 

279 

918 

284 

289 

293 

298 

303 

308 

312 

3*7 

3-22 

327 

•7   3-5 

81  rv 

919 

33i 

336 

34i 

345 

350 

355 

360 

364 

369 

374 

4>o 
•9   4-5 

920 

379 

383  i  388 

393 

397 

402 

407 

412 

4i6 

421 

921 

426 

436 

435 

440 

445 

449 

454 

459 

463 

46§ 

922 

473 

478 

482 

487 

492 

496 

5oi 

506 

511 

51! 

923 

520 

525 

529 

534 

539 

543 

548 

553 

558 

562 

924 

567 

572 

576 

58! 

586 

590 

595 

600 

605 

609 

925 

614 

619 

623 

623 

633 

637 

642 

647 

65?  656 

926 

66  1 

666 

670 

675 

680 

684 

689 

694 

698 

703 

927 

708 

712 

7i7 

722 

72g 

73i 

736 

74i 

745  750 

928 

755 

759 

764 

769 

773 

778 

783 

787 

792  797 

1929 

80  1 

806 

811 

8i5 

826 

825 

829 

834 

839 

843 

980 

848 

853 

857 

862 

867 

87! 

876 

881 

885 

896 

B 

93i 

895 

899 

904 

909 

9i3 

9'8 

923 

927 

932 

937 

4 

T     rv  ? 

932 

941 

946  95  T 

955 

960 

965 

969 

974 

979 

983 

.  I     O.4 
2    O  Q 

933 

488 

993 

997 

*002 

*oo7 

*oii 

*oi6 

*020 

*025 

^030 

* 

•3   1-3 

|934 

97  °34 

°39 

044 

048 

053 

058 

062 

067 

072 

076 

_    Q 

935 

081 

086 

096 

095 

099 

104 

109 

"3 

"8 

123 

.4    1.8 
e    22 

936 

127 

132 

i37 

141 

146 

J51 

'55 

160 

164 

169 

•  3     A  *  * 

.6   2.7 

937 

174 

178 

183 

188 

192 

197 

202 

205 

211 

215 

938 

220 

225 

229 

234 

239 

243 

248 

252 

257 

262 

•7   3-i 

8-  f\ 

939 

26g 

271 

276 

286 

285 

289 

294 

299 

3°3 

308 

j  •  ^ 
.9   4.6 

940 

313 

3i7 

322 

326 

33? 

336 

340 

345 

349  354 

941 

359 

363 

368 

373 

377 

382 

38g 

391 

396 

406 

942 

405 

409 

414 

419 

423 

428 

432 

437 

442 

446 

943 

45' 

456 

466 

465 

469 

474 

479 

483 

488 

492 

944 

497 

502 

5°6 

511 

51! 

520 

525 

529 

534 

538 

^945 

543 

548 

552 

557 

56? 

566 

57o 

575 

580 

584 

1 

1946 

589 

593 

598 

603 

607 

612 

6ig 

621 

626 

636 

947 

635 

639 

644 

649 

653 

658 

662 

667 

671 

676 

948 

681 

685 

690 

694 

699 

703 

7o§ 

7i3 

717 

.722 

949 

72g 

73i 

736 

746 

745 

749 

754 

758 

763 

768 

950 

772 

777 

781 

786 

790 

795 

800 

804 

809 

813 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P. 

341 


TABLE    V.— LOGARITHMS    OF    NUMBERS. 


N. 

0 

1 

2  |  3  |  4 

,"> 

6 

7 

8 

9 

P.  P. 

950 

97  772 

777 

781 

786 

796 

795 

800 

804 

809 

813 

95  1 

818 

822 

827 

83  ? 

836 

841 

845 

850 

854 

859 

952 

863 

863 

873 

877 

882 

88g 

891 

895 

900 

904 

|  953 

909 

914 

9*8 

923 

927 

932 

936 

941 

945 

95o 

!  954 

955 

959 

964 

968 

973 

977 

982 

98g 

991 

996 

1  955 

98  006 

005 

009 

014 

OI§ 

023 

027 

032 

°36 

041 

5 

956 

046 

056 

°55 

°59 

064 

063 

073 

077 

082 

085 

.1 

o.5 

!  957 

091 

095 

100 

i°5 

I0§ 

114 

n§ 

123 

127 

132 

.2 

I.O 

!  958 

136 

141 

J45 

150 

154 

159 

163 

i6§ 

173 

177 

•3 

I-S 

!  959 

182 

185 

191 

195 

200 

204 

209 

213 

218 

222 

•4 

2.0 

|  960 

227 

231 

236 

240 

245 

249 

254 

259 

263 

268 

2.5 

961 

272 

277 

281 

286 

290 

295 

299 

3°4 

3°8 

313 

3-o 

962 

3*? 

322 

326 

33i 

335 

340 

344 

349 

353 

358 

•7 

3-5 

;  963 

362 

367 

37i 

376 

386 

385 

38§ 

394 

398 

403 

.8 

4.0 

i  964 

40? 

412 

4ig 

421 

425 

43° 

434 

439 

443 

448 

•9 

4-5 

!  965 

452 

457 

461 

466 

475 

475 

479 

484 

483 

493 

966 

49? 

502 

506 

511 

5i5 

520 

524 

529 

533 

538 

967 

542 

547 

55? 

556 

566 

565 

56§ 

574 

578 

583 

968 

587 

592 

596 

60  1 

605 

610 

614 

619 

623 

628 

969 

632 

637 

641 

646 

656 

655 

659 

663 

668 

672 

970 

677 

681 

686 

690 

695 

699 

7°4 

7°8 

7i3 

717 

,  4 

971 

722 

726 

73i 

735 

740 

744 

749 

753 

757 

762 

.1 

0.4 

972 

76g 

771 

775 

780 

784 

789 

793 

798 

802 

807 

.2 

°-5 

973 

811 

8i5 

820 

824 

829 

833 

838 

842 

847 

85? 

•  3 

i-3 

974 

856 

865 

865 

869 

873 

878 

882 

887 

891 

896 

•4 

1.8 

975 

906 

9°5 

9°9 

914 

9J8 

922 

927 

93i 

936 

940 

•  5 

2.2 

976 

945 

949 

954 

958 

963 

967 

971 

976 

980 

985 

.6 

2.7 

977 

989 

994 

998 

*oo3 

*oo7 

Oil 

*oi6 

*020 

*025 

*O29 

•  7 

3-1 

978 

99  °34 

°38 

043 

047 

°5? 

056 

066 

065 

069 

074 

.8 

3-5 

979 

078 

082 

087 

091 

096 

100 

i°5 

109 

113 

118 

-9 

4.0 

980 

122 

127 

131 

136 

146 

*45 

149 

153 

158 

162 

981 

I67 

171 

176 

1  80 

184 

189 

193 

198 

202 

2og 

982 

211 

215 

220 

224 

229 

233 

237 

242 

246 

251 

983 

255 

260 

264 

26§ 

273 

277 

282 

28g 

296 

295 

984 

299 

304 

3o§ 

312 

317 

321 

326 

330 

335 

339 

985 

343 

348 

352 

357 

361 

365 

370 

374 

379 

383 

4 

986 

387 

392 

396 

401 

405 

409 

414 

4i§ 

423 

427 

.1 

0.4 

987 

43i 

436 

446 

445 

449 

453 

458 

462 

467 

47i 

.2 

0.8 

988 

475 

480 

484 

489 

493 

497 

502 

506 

5" 

515 

•  3 

1.2 

989 

5i9 

524 

528 

533 

537 

54i 

546 

55o 

554 

559 

•4 

1.6 

990 

563 

568 

572 

576 

58i 

585 

59° 

594 

598 

603 

•  5 

2.0 

991 

607 

611 

616 

626 

625 

629 

633 

638 

642 

647 

.  6 

2.4 

992 

651 

655 

660 

664 

66g 

673 

677 

682 

686 

696 

•  7 

2.8 

993 

695 

699 

703 

708 

712 

717 

721 

72j 

730 

734 

.8 

3-2 

994 

738 

743 

747 

75? 

756 

766 

765 

769 

773 

778 

•9 

3.6 

|  995 

782 

78g 

791 

795 

800 

804 

803 

813 

817 

821 

996 

826 

836 

834 

839 

843 

847 

852 

856 

86  1 

865 

997 

869 

874 

878 

882 

887 

891 

895 

900 

904 

9°8 

998 

9i3 

91? 

922 

926 

930 

935 

939 

943 

948 

952 

999 

956 

961 

965 

969 

974 

978 

982 

987 

991 

995 

1000 

00   000 

004 

oog 

013 

017 

021 

026 

036 

°34 

°39 

i  N- 

O 

1 

2 

3 

4 

^) 

6 

7 

8 

9 

P.P. 

342 


TABLE   V.— LOGARITHMS    OF   NUMBERS. 


N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P 

0 

1000 

ooo  ooo 

043 

087 

136 

i73 

217 

266 

3°4 

347 

390 

01 

02 
03 
04 

°5 
06 

07 
08 
09 

434 
867 
ooi  301 

733 
002  166 

598 
003  029 
466 
891 

47? 
911 

344 

777 
209 
641 
o72 

503 
934 

521 
954 
387 
820 

684 

546 
977 

564 
997 
43  ! 
863 
295 
727 

*O2O 

607 
^041 
474 
9°8 
339 
776 

202 

*633 

*65' 

95° 
382 
814 

245 
676 

*io8 

694 

*I27 

566 

993 
425 
857 
288 
719 
*T49 

*737 
*i7i 

604 

463 
900 

^62 

647 

5lf 
943 

374 
805 

824 
696 

555 
988 

848 

.1 

.2 

•  3 

81 
13.6 

43 

11 

12.9  J 

1010 

004  32! 

364 

407 

45° 

493 

536 

579 

622 

665 

708 

•4 

174 

I7.2 

ii 

12 
14 

005  i  86 
609 
006  038 
466 

794 
223 

081 
509 

837 
265 
695 
123 

880 

309 
738 

594 

923 
352 

209 

637 

966 

395 
824 

252 
680 

438 
865 

295 

722 

481 
909 

337 

*094 

523 
952 
386 
808 

995 
423 
851 

•9 

5:1 

39^ 

2I-5 
25.8 

30-1 
34-4 
38.7 

16 

17 
18 

19 

893 
007  321 
748 
008  174 

936 

796 
217 

979 
406 
833 
259 

*O22 

449 

875 
302 

491 

9J8 
344 

534 
387 

*oo;$ 
430 

*i93 
620 

*046 

472 

662 
*o89 
5'5 

705 
55? 

1020 

600 

642 

685 

728 

770 

813 

855 

898 

946 

983 

21 
22 
23 
24 

25 
26 

27 
28 
29 

009  025 

8rS 

oio  300 
724 
on  147 

570 
993 

OI2  415 

003 

493 
918 

342 
766 
189 
612 

45? 

in 

536 
966 

385 
803 
232 

655 
*o77 
500 

153 

*57§ 
*oo3 

427 

851 

274 

697 

*I2O 

542 

196 
621 

469 
893 
316 

739 
*i62 

584 

238 
663 

*o88 

512 
935 
359 

J82 

281 
J06 

554 
978 
401 
824 
*248 

663 

323 
748 

*I72 

598 

*O20 

443 
*86§ 
716 

366 
796 

*2I5 
/3S 
486 

753 

4°8 
^33 

681 
*IO5 
528 

*951 
373 

795 

.  i 

.2 

•3 
•  4 

•7 
.8 
•9 

a 

17.0 

21.2 

25-5 

29.7 

42 

4.2 

16.8 

21.  0 
25.2 

29.4 

33-6 
37-8 

1030 

837 

8/9 

921 

963 

*oo6 

*o48 

*09o 

*I32 

!74 

218 

32 
33 
34 
35 
36 

37 
38 
39 

013  258 
679 

014  106 

526 
946 
015  360 

779 
016  197 

301 

722 
142 

562 
982 
401 
826 
239 
657 

343 
764 
184 
604 

443 
862 
281 
699 

385 
806 

646 

*o66 
485 
904 

323 

74i 

427 
848 
263 
683 
*io8 
527 
946 

364 
782 

469 
890 

*735 

56§ 
988 

406 
824 

932 

352 
772 

*I92 

611 

448 
865 

553 
974 
394 
814 

653 

*072 

490 
908 

595 
*oi6 

438 
*85g 
695 

532 
950 

637 

*°58 
478 

898 
*3'8 

737 

573 
991 

.1 

.2 

•3 

.4 

4-i 

12.4 
16.6 

41 

4.1 

8.2 

12.3 
16.4 

1040 

017  033 

°75 

117 

'58 

200 

242 

284 

325 

367 

409 

•  5 
.6 

20.7 
24.9 

20.5 
24.6 

42 
43 
44 
45 
46 

47 
48 

49 

867 
018  284 
706 
019  ng 

946 
020  361 

775 

492 
909 
326 

742 
158 

573 
988 
402 
817 

534 
95^ 

367 

783 
199 
614 

444 
858 

576 
992 
409 

825 
241 
656 

485 
899 

617 

*034 

867 
282 
697 

*II2 

527 
941 

659 
*o76 

492 

9°8 
324 
739 
*i54 
563 
982 

534 

95° 
365 
786 

610 

*024 

*<51 
575 
991 
407 
822 

*237 
65? 
*o65 

784 

*2OI 
617 

*033 

448 
863 

692 

*io8 

826 

*242 
659 

49° 

905 

*320 

734 
*i48 

-9 

29.6 
33-2 
37-3 

28.7 
32.8 
36.9 

1050 

021  189 

236 

272 

3*3 

354 

396  ! 

437 

478 

520 

561 

N. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.P 

343 


TABLE   V.— LOGARITHMS    OF    NUMBERS. 


i  N. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

|1050 

021  189 

230 

272 

3J5 

3V4 

396 

437 

478 

520 

56? 

51 
52 
53 
54 
55 
56 

57 
58 
59 

602 
022  OlJ 

428 

846 
023  252 
664 
024  075 

485 
896 

644 

057 
469 

882 
293 
705 
116 

52§ 
937 

685 
°98 
511 
923 
335 
746 

157 
568 

978 

726 
139 
552 
964 
376 
787 
198 
609 
*oi9 

»;68 
181 
593 
*oo<5 

4i7 

823 

*39 
650 
*o6o 

809 

222 
634 

*°46 
458 
869 

280 
691 

*IOI 

856 
263 
676 
*o88 

499 
916 

321 
,732 

*I42 

892 

304 
717 

*I29 

540 
95! 
362 

773 
*iS3 

933 
346 

758 

*I70 

58? 
993 

403 
814 

*224 

974 
387 
799 

*2II 

/23 

034 

444 
855 
*265 

2 

3 
4 

I 

9 

41 

K 

12.4 

16.6 
20.7 
24.9 

29.0 
33-2 
37-3 

1060 

025  306 

347 

388 

429 

469 

$10 

551 

592 

633 

674 

61 
62 
63 
64 

65 

66 

67 
68 
69 

715 
026  124 

533 
941 
027  349 

757 
028  164 
57? 
977 

756 
165 
574 
982 

39° 
798 

205 
612 

*oi§ 

797 

20g 

6i5 

*023 

431 

838 
246 
652 
*°59 

838 

247 
656 

*o64 

472 
879 

285 
693 
*°99 

879 

288 

696 
*I05 
5l2 
920 

327 

734 
*i4o 

920 
329 
737 
*i4§ 

553 
961 

368 

774 
*i8i 

961 

370 

778 
*i8g 
594 
*ooi 

4°8 

*8I5 

*22I 

*002 
4IO 
819 

*22? 

*635 
*042 

449 
856 

*262 

*O42 

451 
860 

*268 

675 
*o83 

490 
896 

*302 

*o83 

495 
901 

*3o9 

*7I§ 
I23 

530 

*937 
*343 

.1 

.2 

•3 
•4 

:I 

•  7 
.8 
•9 

41 

4.1 

8.2 

12.3 
16.4 

20.5 

24.6 

28.7 
32.8 

36.9 

1070 

029  384 

424 

465 

5°5 

546 

58g 

627 

668 

7°8 

749 

7i 

72 

73 

i   74 
75 
76 

77 
78 

I   79 

789 
030  195 

599 
031  004 

4°8 
812 

032  215 
619 

033  021 

830 

23! 
640 

044 

449 

852 

256 

659 
061 

876 
276 
685 
085 
489 
893 

296 
699 

102 

911 

3i§ 
721 

"| 

529 
933 
336 
739 
142 

95i 

357 
761 

166 
57o 
973 

377 
780 
182 

992 

397 
802 

20§ 

610 
*oi4 

4i? 

820 

222 

*032 

438 
842 

247 
*65i 

°54 

457 
866 
263 

*°73 
478 
883 

287 
691 
*°94 

498 
906 

303 

*n4 
519 
923 
32? 

*73f 
*i35 

538 
941 

343 

*i54 
559 
964 

368 
772 
*i75 

578 
981 

383 

i 

2 

3 
4 

I 

9 

46 

4.0 

8.1 

12.  I 

ID.  2 
20.2 
24.3 

28.3 
32-4 
36.4 

11080 

424 

464 

504 

544 

584 

625 

665 

705 

745 

78! 

81 
82 
83 
84 
85 
86 

87 
88 

89 

825 
034  227 
62§ 
035  02§ 
42§ 
830 
036  22§ 
629 
037  028 

866 
267 
66§ 
069 
470 
870 
269 
669 
068 

906 

307 

7°8 
109 

510 
910 

3°9 
7°8 
107 

946 
347 
748 
149 
55° 
95° 
349 
748 
14? 

986 

388 

789 
189 
59° 
99° 
389 
78§ 
187 

*02g 
428 
829 
22§ 
630 
*02§ 

42§ 

82§ 
227 

*o6g 
468 
869 
269 
670 
*o6§ 
469 
863 
267 

*io7 
508 
909 

3°9 

*71? 
^109 

509 

908 

307 

H7 
548 
949 
349 

*75? 
*i49 

549 
948 

347 

187 
58§ 
989 
389 
79° 
*iSc> 

589 
988 

386 

i 

2 

3 
4 

I 

9 

40 

4-0 

8.0 

12.  O 

16.0 

20.0 
24.0 

28.0 
32.0 
36.0 

1090 

426 

46g 

506 

546 

586 

625 

665 

705 

745 

785 

91 
92 

93 
94 

95 
96 

97 
98 

99 

825 

038  222 
620 
039  017 
414 

816 

040  20§ 
602 

997 

864 
262 
660 

057 
454 
850 

246 
642 

*°37 

904 
302 
699 

°96 
493 
890 

286 
681 

*°76 

944 
342 
739 
136 
533 
929 

32! 
721 
*n6 

984 
38! 
779 
176 

575 
969 

365 
766 

*i55 

*023 
421 
819 
216 
612 

*00§ 

404 
800 

*!95 

*o6j 
461 

858 

255 
652 

*048 

444 
839 
*234 

*io3 

501 
898 

295 
691 
*o88 

483 
879 
*274 

143 
540 
938 

335 
73i 

*I27 

523 
9*8 
*3*3 

183 
586 

977 
374 
771 
*i67 

563 
958 
*353 

i 

2 

3 
4 

I 
I 

9 

35 

3-5 

7-9 
n-8 

15-8 
19  7 
23-7 

27-6 
31.6 

35-5 

1100 

041  392 

432 

47? 

51* 

55° 

59° 

629 

669 

7°8 

748 

N. 

O 

1 

2 

3 

)f^l 

5 

6 

7 

8 

9 

P. 

P. 

344 


TABLE   VI.— LOGARITHMIC    SINES    AND    TANGENTS    OF   SMALL   ANGLES. 


log  sin  0  =  log  0"  +  S.              O°              lo&  ^  '  ~  lo£  sin  ^  +  'S"' 
log  tan  0  =  log  0"  -\-  T.                              log  0"  =  log  tan  0  -f  7". 

" 

' 

S 

T 

Log.  Sin. 

S' 

T 

Log.  Tan.  | 

o 
60 

120 
1  80 

240 

0 

I 

2 

3 

4 

4.685  57 

1 

! 

OO 

6.46  372 

.76475 
.94084 
7.06  578 

5-3H42 
42 
42 
42 
42 

42 
42 
42 
42 
42 

—  00 

6.46  372 

.76475 
.94  084 
7.06  578 

300 
360 
420 
480 
540 

6 

8 
9 

4-685  5? 

1 

i 

5? 

7.16  269 
.2418? 
.30882 
.36681 
41  797 

5.3H42 
42 
42 
42 
42 

42 
42 
42 
4§ 
42 

7.16269 
.24  1  88 
.30  882 
.36681 
•4i  797 

600 
660 
720 
780 
840 

10 

ii 

12 
13 
H 

4-685  5? 

1 

I 

746  372 
.50512 
.54296 
•57  767 
.60985 

5.31442 
42 
42 
42 

42 

42 
42 
42 
42 
42 

746  372 
.50512 
.54  291 
.57  767 
.60  985 

900 

;  96o 

*020 
1080 
I  I4O 

15 

16 

11 

19 

1 

57 

58 
58 
58 
58 
58 

7-63  981 
.66  784 

•6941? 
.71  899 
^.74.248 

5.31442 
42 
42 
42 
42 

42 
42 
42 
42 
42 

7.63  982 
.66785 
.69418 
.71900 
•74  248 

1200 
1260 
1320 
1380 
1   1440 

20 

21 
22 

23 

24 

4.685  57 
57 
57 
57 
57 

58 
58 
58 
58 
58 

776475 
.78  594 
.80614 
.82  545 
.84  393 

5.31443 
43 
43 
43 
43 

42 
42 
42 
42 
42 

776  476 
.78  595 
.80615 
.82  546 
.84  394 

1500 
1560 
1620 
1680 
1740 

§ 

27 
28 
29 

4.685  57 
57 
57 
57 

57 

58 

11 

58 
58 

7.86  1  66 
.87  869 

•89  508 
.91  088 
.92612 

5-31443 
43 
43 
43 
43 

4i 
41 

41 

4? 
4i 

7.86  1  6? 
.87871 
.89  510 
.91  089 
.92613 

1800 

1860 
1920 
1980 
2040 

30 

3i 

32 
33 
34 

4.685  57 
57 
57 
57 

57 

58 
58 
58 
59 
59 

7.94  084 
.95  508 
.96887 
.98  223 
•99  520 

5-3H43 
43 
43 
43 
43 

4? 

4! 

41 
4i 
4i 

7.94086 
.95510 
.96889 

.98  225 
•99  522 

2100 
2l6o 
2220 
2280 
2340 

3 

% 

39 

4.685  56 

i 

56 

59 
59 
59 
59 
59 

8.00  778 

.02  002 
.03  192 

.04  35° 
.05  478 

5.3H43 
43 
43 
43 
43 

4i 
4i 
4i 
40 
46 

8.00781 
.02004 
.03  194 
.04352 
.05  481 

2400 
2460 
2520 
2580 
2640 

40 

4i 
42 
43 
44 

4.685  56 
56 

It 

59 
59 
59 
60 
60 

8.06  577 
.07  650 
.08  695 
.09713 
.10716 

5.3H43 
43 
43 
43 

43 

40 
46 
46 
40 
40 

8.06  586 
.07  653 
.08  699 
.09  721 
.10726 

2700 
2760 
2820 
2880 
2940 

4I 
46 

47 
48 

49 

4.685  56 

5^ 
56 

56 

56 

60 
60 
60 
66 
66 

8.  1  1  692 
.12647 
.13581 
.14495 
.15390 

5.3H44 
44 
44 
44 
44 

40 
40 
40 

3? 
39 

8.11696 

.12  651 

'13585 
,14499 

•15395 

3000 
3060 
3120 
3180 

3240 

50 

5i 

52 
53 
54 

4.685  56 

1 

it 

66 
66 
61 
61 
61 

8.16268 
.17  128 
.1797* 

.18798 
.19610 

5.3H44 
44 
44 
44 
44 

39 
39 
39 
39 
39 

8.16272 
.17133 

•17976 
.18803 

•19615 

3300 
3360 
3420 
3480 
3540 

55 
56 

% 

59 

4.685  55 
55 

i 

y 

61 
61 
61 
61 
62 

8.20  407 

.21  l8§ 
.21  958 
.22713 

.23  415 

5-3H44 
44 
44 
44 
44 

39 
38 
38 

i 

8.20412 

.21  I95 
.21064 
.22719 
.23462 

345 


TABLE   VI.— LOGARITHMIC    SINES    AND    TANGENTS    OF   SMALL   ANGLES. 


log  sin  0  =  log  0'^-f-  &               1  °              log  $"  —  l°g  sin  0  H~  S'. 
log  tan  0  =  log  0"  -f-  T.               *               log  0"  —  log  tan  0  -f-  7". 

// 

/ 

S 

T 

Log.  Sin. 

S' 

T 

Log.  Tan. 

3600 
3660 
3720 
3780 
3840 

0 

I 

2 

3 

4 

4.685  55 
55 
55 
55 

55 

62 

62 
62 
62 
62 

8.24  185 
.24  903 
.25609 
.26  304 
.26  988 

5.31444 
45 
45 
45 
45 

38 
38 

1 

8.24  192 
.24910 
.25615 
.26311 
.26  995 

3900 
3960 
4020 
4080 
4140 

6 

8 
9 

4.685  55 

I 

54 

62 
63 
63 
63 
63 

8.27  66T 
.28  324 
.28  97? 
.29  620 
.30254 

5-3H45 
45 
45 
45 
45 

3? 
37 
37 

I 

8.27  669 
.28  332 
.28  985   \ 
.29  629 
.30  263 

4200 
4260 
4320 
4380 
4440 

10 

ii 

12 
13 
14 

4.685  5$ 
54 
54 
54 
•   54 

63 
63 
64 
64 
64 

8.30  879 
•31  495 

.32  102 

.32  701 

•33  292 

5.31445 

45 
45 

46 
46 

II 
? 

36 

8.30  88§ 
.31  5°4 

.32  112 
.32711 
•33  302 

4500 
4560 
4620 
4680 
4740 

15 

16 

17 
18 

19 

4.685  54 
54 

54 

a 

64 
64 
65 
65 
65 

8.33875 
•34  456 
.35018 

•35  578 
.36  131 

5.31446 

46 

46 
46 

46 

11 

35 
35 
35 

8.33  885 
.34461 
.35  029 
.35  58§ 
.36  143 

4800 
4860 
4920 
4980 
5040 

20 

21 

22 

23 
24 

4.685  53 
53 
53 
53 

53 

65 
65 

65 
66 
66 

8.3667? 
.37217 

•37  75o 
.38  276 
.38  796 

5-31446 
46 
46 
4& 
47 

34 

3| 
34 
34 
34 

8.36  689 
•37  229 
.37  762 
.38  289 
.38  809   , 

5100 
$160 
5220 
5280 
5340 

3 

27 
28 
29 

4-685  53 
53 

11 

52 

67 
67 
67 

8.39310 
.39818 
.40  320 
.40816 
41  307 

5.31447 
47 
47 
4? 
4? 

33 
33 
33 
33 
33 

8-39  323 
•39  83? 
40  334 
.40  836 

4i  32?^ 

5400 
5460 
5520 
5580 
5640  . 

30 

3i 
32 
33 
34 

4.685  52 
52 
52 
52 
52 

6J 
6? 
68 
68 

68 

8.41  792 
.42  271 
.42  746 
43215 
43  680 

5'3!44? 
4? 
4? 
48 

48 

32 
32 
32 
32 
3i 

8.41  807 
.42  287 
.42  762 
43  231 
43696 

5700 
5760 
5820 
5880 
5940 

% 

% 

39 

4.685  52 
52 

1 

68 
69 

i 

69 

6§ 

844  139 
44  594 
45  °44 
45  489 
45  930 

5.31448 
4? 

4§ 

3i 
31 

31 
30 

30 

8.44  156 
.44611 
.45  o6f 
45  507 
45  948 

6000 
6060 
6l20 
6180 
6240 

40 

4i 

42 

43 
44 

4.685  51 
5i 
5i 
5i 
5i 

6§ 
70 

7? 
70 

70 

8.46  365 
46  798 

47  22g 

47  650 
.48  069 

5.3H  48 
49 
49 
49 

49 

30 
30 
30 
29 
29 

846  385 
.46817 

47  245 
47  669 
.48  089 

6300 
6360 
6420 
6480 
6540 

45 
46 

47 
48 

49 

4.685  56 

5° 
56 
56 
5o 

7i 

K 

72 
72 

8.48  485 
48  89§ 
49  304 
.49  7o8 
.50  108 

5.3I449 
49 
49 
49 
50 

29 

28 

28 

28 
28 

8.48  505 
.48  917 
49  325 
49  729 
.50130 

6600 
6660 
6720 
6780 
6840 

50 

5i 

52 
53 
54 

4.685  50 
5o 

5? 

49 
49 

9 

73 
73 

73 

8.50  504:  - 
.50  89? 
.51  28§ 
.51672 
.52055 

5.3H50 
50 
50 
50 
5o 

2? 

1? 
27 

27 
26 

8.50  526 
.50  920 

•51  3*0 

.51  696 
.52  079 

6900 
6960 
7020 
7080 
7140 

P 

8 

59 

4.685  49 
49 
49 
49 
49 

73 
74 
74 
74 
75 

8.52434 
.52  810 
•53183 
•53  552 
•53918 

5.31453 
5i 
5i 
5i 
51 

26 

26 
25 
25 
25 

8.52453 
.52835 
•53  208 
.53  578 
•53944 

346 


TABLE   VI.— LOGARITHMIC    SINES    AND    TANGENTS    OF   SMALL   ANGLES. 


log  sin  0  =  log  0r'  -|-  St              oo              log  0"  =  log  sin  0  -f-  5'. 
log  tan  0  =  log  0"  -f-  7".              ^               log  0"  =  log  tan  0  -j-  7". 

// 

7200 
7260 
7320 
7380 
7440 

/ 

S 

T 

Log.  Sin. 

S' 

T 

Log.  Tan.  i 

0 

i 

2 

3 
4 

4-685  48 

48 
48 
48 

75 
73 

76 

76 

8.54  282 
.54642 
•54999 
•55  354 
•55  7o5 

5.31451 

5i 
52 
52 

25 

2^ 
24 
24 
23 

8-54  308 
.54  669 
.55027 
.55  381 

-55733 

7500 
7560 
7620 
7680 
7740 

J 

8 
9 

4.685  48 
48 
47 

4? 
47 

76 

8 

7? 
78 

8.56  054 
.56400 
.56  743 
•57  083 
.57421 

5-3I452 
52 

1 

23 
23 

22 
22 

22 

8.56083 
.56429 
.56  772 

•57II3 

-57452 

7800 
7860 
7920 
7980 
8040 

10 

ii 

12 
13 
14 

4.685  47 
47 
47 
46 
46 

78 
78 
79 
79 
79 

8.57  756 
.58089 
.58419 

•58  747 
.59072 

5.3H53 
53 
53 
53 
53 

22 
21 
21 
21 
20 

8-57  787 

.58  121 
.5845? 

-58  779 
•59  105 

8100 
8160 
8220 
8280 
8340 

11 

17 

18 
19 

4-68545 
46 
46 
46 
45 

80 
80 
86 
81 
81 

8-59  395 
•59715 
.60033 
.60349 
.60662 

5-3I453 
54 
54 

54 

54 

20 
20 
19 
19 
»9 

8.59423 

•59749 
.6006? 
.60384 
.60  698   . 

8400 
8460 
8520 
8580 
8640 

20 

21 
22 

23 
24 

4.68545 
45 
45 
45 
45 

81 
82 
82 
82 
83 

8.60  973 
.61  282 
.61  589 
.61  893 
.62  196 

5.3H  54 
54 
55 
55 

55 

18 
if 
17 

8.61009 
.61  319 
.61  626 
.61  931 
.62  234 

8700 
8760 
8820 
8880 
8940 

a 

27 
28 
29 

4.68544 
44 
44 
44 

44 

83 
83 
84 
84 
84 

8.62496 
.62  795 
•63091 

•63  385 
.63  67? 

5-3I455 

y 

I 

16 

'6 

16 
ij 
l| 

8.62  535 
.62  834 
.63  131 
.63425 
•63718 

9000 
9060 
9120 
9180 

;  9240 

30 

3i 
32 
33 

34 

4.685  43 
43 
43 

43 
43 

85 
8£ 
86 
86 
86 

8.63  968 
.64  255 
.64  543 
.64  82? 
.65  1  10 

5-3H  56 

56 
57 
57 

11 

14 

14 
i§ 

0X64  009 
.64  298 

.64585 
.64  876 
.65153 

9300 
9360 
9420 
9480 
9540 

i 

11 

39 

4.685  43 
42 
42 
42 
42 

87 
8? 
g 

00 

OO 

88 

8.65  391 
.65  670 

.65  947 
.66  223 
.66  497 

,3.457 
| 

58 

13 

12 
12 
12 
II 

8.65  435 
.65715 

•65993 
.66269 

•66543 

9600 
9660 
9720 
9780 
9840 

40 

4i 
42 

43 
44 

4.685  42 
4l 

4? 

4i 
4i 

89 
89 

89 
90 

96 

8.66  769 
.67  039 
.67  308 

•67  575 
.67  846 

5-3I458 

58 
58 
59 
59 

II 
10 
10 
10 

09 

8.66816 
.67  087 

•67356 
.67  624 
.67890 

9900 
9960 

10020 
10080 
IOI40 

45 
46 

47 
48 

49 

4.685  41 
40 
40 
40 
40 

91 
9i 
9i 
92 

92 

8.68  104 
.68  36£ 
.68  627 
.68  88§ 
.69144 

5-3I459 
59 
59 
60 
60 

09 

o§ 
08 
08 

of 

8.68154 
.68417 
.68673 
.68  938 
.69196 

IO200 

!  10260 

IO32O 
10380 
10440 

50 

5i 

52 
53 
54 

4.685  40 

i 

39 
39 

93 
93 
93 
94 
94 

8.69  400 
.69654 
.6990? 
.70159 
.70409 

5.31460 
66 
66 
61 
61 

07 

06 
o5 

8.69453 
.69  708 
.69961 
.70214 
.70  464 

10500 
10560 
I062O 
10680 
10740 

P 
% 

-J>9 

4.685  3§ 
38 
38 
38 
38 

95 

95 
96 

96 
97 

8.70  65? 
.70905 
.71  156 

•7i  395 

.71633 

5.31461 
61 
6! 
62 
62 

05 
04 

04 
03 

03 

8.70714 
.70962 

.71  20§ 

•71  453 
.71697 

347 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

0° 


/ 

Log.  Sin. 

D 

Log.  Tan. 

Com.  D. 

Log.  Cot. 

Log.  Cos. 

0 

I 

2 

3 

4 

00 

6.46  372 

6.76475 
6.9408^ 

7.06  57§ 

30103 
17609 
12494 

nfinr 

—  oo 
6.46  372 
6.76475 
6.94083 
7.06  578 

30103 
17609 
12494 

-J-   00 

3.5362? 
3.23  523 
3.05915 
2.93  421 

0.00000 

o.ooooo 

0.  00  000 

o.ooooo 

0.00000 

GO 

59 
58 

I 

I 

7 

8 

9 

7.16269 
7.24  1  8? 
7.30882 
7.3668? 
7.41  797 

9091 
79l8 
6695 

5799 
S»5 

A--S 

7.16269 
7.24  1  88 
7.30882 
7.3668? 
7.41  797 

9691 
79*8 
6694 
5799 
5"3 

2.83730 
2.75812 
2.6911? 
2.63313 
2.58203 

0.00000 
0.00000 

o.oo  ooo 

O.OO  000 

o.ooooo 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 

H 

7.46  372 
7.50512 
7.54290 
7.57767 
7.60985 

4139 
3778 
3476 
32I8 

7.46  372 
7.50512 
7.54291 
7.57767 
7.60985 

4575 
4139 
3779 
3476 
32I8 

2.5362? 

2.49488 

2.45  709 
2.42  233 

2.39013 

0.00  000 
0.00000 

9-99999 

9.99999 
9.99999 

50 

49 

48 

47 

46  ! 

15 

16 

17 
18 

19 

7-6398! 
7.66784: 
7-694i? 
7.71  899 

7.74248 

2990 
2803 
2633 
2482 
2348 

7.63982 
7.66785 
7.69418 
7.71906 

7-74248 

2996 
2803 
2633 
2482 
2348 

2.36018 

2.33215 
2.30  582 
2.28099 
2.25751 

9.99999 
9.99999 

9-99999 
9.99999 

9-99999 

45 

44 
43 
42 
4i 

20 

21 

22 

23 

24 

7.76475 
7.78  593 
7.80613 
7.82545 
7.84393 

2119 

2020 
1930 

1843 

7.76476 
7-78  595 
7.80615 
7.82  546 
7.84393 

2119 

2020 
1930 
I84§ 

2.23  524 

2.21  405 
2.19383 
2.17454 
2.15605 

9-99999 
9-99999 
9.99999 

9-99999 
9.99999 

40 

39 
38 

1 

25 
26 
27 
28 
29 

7.86  1  66 
7.87  869 
7.89503 
7.91  088 
7.92612 

1772 

1703 
1639 

1579 
1524 

7.86  1  6? 
7.87871 
7.89510 
7.91  089 
7-92613 

X773 
I70§ 
1639 

1579 
1524 

IA-72 

2.13832 
2.12  129 
2.10490 
2.08  916 

2.07  38§ 

9-99999 
9.99999 

9-99998 
9-99998 
9-99998 

35 
34 
33 
32 
3i 

30 

31 
32 
33 
34 

7.94084 
7.95  508 
7.96  887 
7.98  223 
7.99  520 

1424 

1379 
1336* 
1296 

12cS 

7.94086 
7.95  510 
7.96889 
7.98  225 
7-99  522 

1424 
1379 

1336 

I29§ 

2.05914 
2.04  490 
2.03  III 

2.01  773 

2.00  478 

9-99998 
9.99998 
9.99998 

9-99  998 
9-99998 

30 

29 
28 
27 
26 

t   P 
% 

39 

8.00773 

8.02  002 
8.03  192 
8.04  350 
8.05  478 

I22§ 
1190 
1158 
XI28 

8.00781 
8.02  003 
8.03  193 
8.04352 
8.05  481 

1223 
1190 
1158 

"28 

I.992I9 

1.97995 
1.96  8o5 
1.95  64? 
I.945I9 

9-9999? 
9-9999? 
9-9999? 
9.9999? 

9-99997 

25 
24 
23 

22 
21 

40 

4i 

42 
43 

!      44 

8.06  57? 
8.07  650 
8.08  69£ 
8.09713 
8.10716 

1072 

1046 

1022 

998 

8.06  586 
8.07  653 
8.08  699 
8.09721 
8.10726 

1072 
1045 

1022 

999 
076 

I.934I9 

1.92  347 
1.91  306 

1.90273 
1.89279 

9-99997 
9-99997 
9-99997 
9-99996 
9-99996 

20 

19 
18 

17 
16 

4I 
46 

47 
48 

49 

8.  1  1  692 
8.12647 
8.13  581 

8.i449§ 
8.15390 

954 
934 
914 

895 
877 

8.11696 

8.12651 
8.13585 
8.14499 
8.15395 

954 
934 
914 

895 

8-77 

1.88  303 
1.87  349 
1.86415 
1.85  506 
1.84605 

9-99996 
9.99996 
9.99996 
9.99996 
9-99995 

15 
H 
13 

12 
II 

50 

5i 

52 
53 
54 

8.16268 
8.17  128 
8.1797? 
8.18798 
8.19610 

860 

843 
827 
811 

8.16272 
8.I7I33 
8.17976 
8.18803 
8.19615 

865 

843 
827 
812 

1-8372? 
1.82867 
1.82023 
1.8!  195 
1.80383 

9-999^5 
9-99995 
9-99995 
9-99995 
9-99994 

10 

9 
8 

6 

\l 
11 

59 

8.20407 
8.21  189 
8.21  958 

8.22713 
8.23455 

797 

782 

76§ 
755 
742 

8.20412 
8.21  195 
8.21  964 

8.22  7I§ 
8.23462 

797 
783 
76§ 
755 
742 

1.7958? 
1.78803 
1.78036 
1.77  280 
1.76538 

9.99993 
9.99994 
9.99994 
9-99994 
9-99993 

5 
4 
3 

2 

60 

8.24  185 

73° 

8.24  192 

73° 

1.75808 

9.99993 

0 

Log.  Cos. 

D 

Log.  Cot. 

Com.  D. 

Log.  Tan. 

Log.  Sin. 

'       \ 

89C 


348 


TABLE  VII.— LOGARITHMIC  SIXES,  COSINES,  TANGENTS,  AND  COTANGENTS. 


/ 

Log.  Sin. 

D 

Log.  Tan. 

Com.  D. 

Log.  Cot. 

Log.  Cos. 

0 

I 

2 

3 

4 

8.24185 
8.24903 
8.25609 
8.26304 
8.26988 

718 
706 
694 
684 

8.24  192 
8.24910 
8.25615 
8.26311 
8.26995 

718 
7°6 
695 
684 

1.75808 
1.75090 
1.74383 
1.73688 
1.73004 

9-99993 

9-99993 
9-99993 
9.99992 
9.99992 

60 

59 
58 

1 

1 

8 
9 

8.27661 
8.28324 

8.28  97? 
8.29626 
8.30254 

673 
663 

653 
643 

634 

8.27669 
8.28332 
8.28985 
8.29629 
8.30263 

673 
663 

653 
643 
634 

1.72331 
I.7I  667 
I.7I  014 
1.70371 

1-69736 

9.99992 

9-99992 
9.99992 
9.99991 
9.99991 

55 
54 
53 
52 
5i 

10 

ii 

13 
14 

8.30879 
8.31495 

8.32  102 
8.32701 
8.33292 

625 
616 
607 
599 
59i 

8.30883 
8.31  504 
8.32  112 
8.3271? 
8.33302 

023 
6l6 
607 

599 
59i 

1.69  ill 
1.68495 
1.67888 
1.67283 
1.66697 

9.99991 
9.99996 
9.99996 
9.99990 
9.99990 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

8.33875 
8.34456 
8.35018 

8-35578 
8.36  131 

583 
575 
567 
5« 
553 

8.33885 
8.34461 
8.35029 
8.35  589 
8.36  143 

583 
575 
568 
560 

553 
,  ,? 

1.66  114 
1.65  539 
1.64971 
1.64416 
1.63857 

9.99989 
9.99989 
9.99989 
9.99989 
9.99983 

45 
44 
43 
42 
4i 

20 

21 

22 

23 
24 

8.36677 
8.37217 
8.37750 
8.38276 
8.38796 

540 

539 
533 
526 
520 

8.36689 
8.37229 
8.37762 
8.38289 
8.38809 

546 

539 
533 
527 
520 

1.63  316 
1.62771 
1.62  238 
1.61  711 
1.61  191 

9.99988 
9.99988 
9.99987 
9.99987 
9.99987 

40 

39 
38 
37 
36 

1  2 

27 
28 
29 

8.39310 
8.39818 
8.40320 
8.40816 
8.41  307 

5*4 

508 
502 
496 
491 

8.39323 
8.39831 

8.40  334 
8.40836 

8.41  321 

5T4 
508 
502 

496 
491 

05 

1.60675 
i.  60  163 
1.59666 
1.59169 
1.58673 

9.99985 
9.99985 
9.99986 
9.99986 
9.99985 

35 
34 
33 
32 
3i 

30 

3i 
32 

33 
34 

8.41  792 
8.42  271 
8.42  746 

843215 
8.43  680 

485 
479 
474 
469 
464 

8.41  807 
8.42  287 
8.42  762 
8.43  231 
8.43  696 

4°5 
480 
475 
469 
464 

A&Q 

1.58193 

I.577I3 
1.57238 
1.56763 
1.56304 

9.99985 
9.99985 
9-99  984 
9.99984 
9.99984 

30 

29 
28 
27 
26 

35 
36 

P 

39 

8.44  139 
8.44  594 
8.45  044 
8.45489 
8.45930 

A-n 

4o9 
454 
450 
44§ 
440 

8.44  156 
8.44611 
8.45  061 
8.45  507 
8.45  948 

45S 
450 

445 
441 

1.55844 
1.55389 
1-54938 
1-54493 
1.54052 

9-99  983 
9-99983 
9.99982 
9.99982 
9.99982 

25 
24 

23 

22 
21 

40 

4i 
42 
43 
44 

8.46  365 
8.46  798 
8.47  225 
8.47650 
8.48  069 

436 
432 
428 
423 
419 

8.46  385 
8.46817 
8.47  245 
8.47  669 
8.48  089 

432 

42§ 

424 
419 

Alfl 

i.536i5 
1.53183 
1.52754 
1.52336 
1.51911 

9.9998! 
9.99981 
9.99981 
9.99986 
9.99980 

20 

19 
18 

17 
16 

45 
46 

47 
48 

49 

8.48485 
8.48  895 
8.49  304 
8.49  708 
8.50108 

4J5 
411 
407 
404 
400 

8.48  505 
8.48917 
8.49325 

8.49729 
8.50  130 

4IO 
4I2 
408 
404 
400 

1.51495 
1.51083 
1.50675 
1.50276 
1.49  870 

9-99979 
9.99979 

9-99979 
9-99978 
9.99978 

15 
14 
13 

12 

II 

50 

5i 
52 
53 
54 

8.  50  504 
8.50897 
8.51285 
8.51672 
8.52055 

396 
393 
389 
386 
382 

8.50525 
8.  50  920 
8.51  310 
8.  5  1  696 
8.52079 

396 
393 
390 
386 

383 
- 

1-49473 
1.49080 
1.48690 
1.48  304 
1.47921 

9.99978 
9-99977 
9-99977 
9-99976 
9-99976 

10 

9    ' 
8 

6 

P 

ii 

59 

8.52434 
8.52810 
8-53183 
8.53552 
8-53918 

379 
375 
373 
36§ 
366 

8-52458 
8.52835 
8.  53  208 
8.53578 
8-53944 

379 
376 
373 
370 
36g 

1.47  541 
1.47  165 
1.46792 
1.46422 
1.46055 

9-99975 
9-99975 
9-99975 
9-99974 
9-99974 

5 

| 

i 

00 

8.54282 

363 

8-54308 

304 

1.45691 

9-99973 

0 

Loir.  Cos. 

D 

Log.  Cot. 

Coin.  1). 

Log.  Tail. 

Log.  Sin. 

/ 

88' 


349 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

2° 


/ 

Log.  Sin. 

D 

Log.  Tan. 

Com.  D. 

Log.  Cot. 

Log.  Cos. 

0 

I 

2 

3 

4 

8.54282 
8.54642 
8.54999 
8-55354 
8.55705 

360 
357 
354 
35? 
348 
346 
343 
340 
338 
335 
332 
335 
327 
325 
323 
320 
318 
3'6 

3!§ 

311 
3°9 
3°6 
3°4 
302 

3°S 
298 
296 
294 
292 
290 
283 
285 
284 
282 
281 
279 
277 
275 
274 
272 
270 

26§ 

267 
265 
264 
262 
260 
259 
257 
256 
254 
253 
251 
250 
248 
247 

245 
244 
243 
241 

8-54308 

8.  54  669 
8.55027 
8.55381 
8-55733 

365 
358 
354 
352 
349 
346 
343 
34i 
338 
335 
333 
333 
328 
325 
323 
323 
3i§ 
3'6 
3M 
3" 
309 
3°7 
305 
3°3 
300 
299 
297 
294 
293 
291 

283 
287 
285 
283 
281 
280 
278 
276 

274 
272 
271 
269 
267 
266 
264 
262 
261 
259 
258 
256 
255 
253 
252 
250 
249 
248 

246 
245 
243 
242 

1.45691 
I-4533I 
1-44973 
1.44618 
I.4426£ 

9-99973 
9-99973 
9.99972 
9.99972 
9.99971 

60 

59 
58 
57 
56 

:       I 

8 
9 

8.56054 

8.  56  400 

8.56743 
8.57083 
8.57421 

8.56083 
8.56429 
8.56772 

8.57H3 

8.57452 

1-43  9J7 
i.43  57i 
i.43  22? 
1.42886 
1.42  548 

9.99971 
9.99971 
9.99976 
9.99970 
9.99969 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 

14 

8-57756 
8.58089 
8.58419 

8.58747 
8.59072 

8.5778? 

8.58  121 
8.5845? 
8.58779 
8.59105 

1.42  212 
I.4I  879 

i.4i  548 

I.4I  220 
1.40895 

9.99969 

9-99  968 
9.99968 
9.9996? 
9.99967 

50 

49 
48 
47 
46 

15 

16 

17 
18 

19 

8-  59  395 
8.59715 
8.60033 
8.60349 
8.60662 

8.59423 

8-59749 
8.6006? 
8.60  384 
8.60698 

I.4057I 
I.4O25I 

L39932 
1.39616 
1.39302 

9.99966 
9.99966 
9.99965 
9.99965 
9.99964 

45 
44 
43 
42 

4i 

20 

21 

22 

23 
24 

8.60973 
8.61  282 
8.61  589 
8.61  893 
8.62  196 

8.  6  1  009 
8.61  319 
8.61  626 
8.61  931 
8.62  234! 

1.38990 
1.38681 

1.38374 
I.38o6§ 
1.37765 

9-99964 
9.99963 
9.99963 
9.99962 
9.99962 

40 

38 

% 

25 
26 

27 
28 

29 

8.62  496 
8.62795 
8.63091 
8.63385 
8.6367? 

8.62  535 
8.62  834 
8.63  131 
8-63425 
8.63718 

1.37465 

1.37  166 
1.36869 

1.36574 
1.36281 

9-99:96i 
9.99961 
9.99966 
9-99959 
9-99959 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 
34 

8.63968 
8.64  255 
8.64  543 
8.6482? 
8.65  no 

8.64  009 
8.64298 
8.64  585 
8.64870 
8.65  153 

1.35996 
1.35702 

I.354H 
1.35  129 

i.3484g 

9-99958 
9.99958 

9-9995? 
9-99957 
9-99956 

30 

29 
28 
27 
26 

Ii 

37 
38 
-   39 

8.65  391 
8.65  670 
8.65  94? 
8.66  223 
8.66497 

8.65435 
8.65715 
8.65  993 
8.66  269 
8.66  543 

1.34565 
1.34285 
1.34007 
I.3373I 

J-33456 

9.99956 
9-99955 
9-99954 
9-99954 
9-99953 

25 
24 
23 

22 
21 

40 

4i 

42 

43 
44 

8.66769 
8.67  039 
8.67  308 
8.67  575 
8.67  846 

8.66816 
8.67087 
8.67  356 
8.67  624 
8.67  890 

1-33184 
1.32913 
1.32643 
1.32376 
1.32  no 

9-99953 
9.99952 

9-99952 
9.99951 
9.99956 

20 

19 

17 
16 

4I 
46 

47 
48 

49 

8.68  104 
8.68  365 
8.68627 
8.68886 
8.69  144 

8.68  154 
8.68417 
8.68678 
8.68938 
8.69  196 

1.31845 
1.31  583 
1.31  321 
1.31  062 
1.30  803 

9.99950 
9-99949 
9-99948 
9.99948 

•     9-9994? 

15 
14 
13 

12 
II 

50 

5i 
52 
53 

54 

8.69400 
8.69654 
8.6990? 
8.70159 
8.  70  409 

8.69453 
8.69708 
8.6996! 
8.70214 
8.  70  464 

I.30547 
1.30  292 
1.30038 
1.29786 

1.29535 

9-99947 
9-99946 
9-99945 
9-99945 
9-99944 

10 

6 

H 
H 

57 

8.7065? 
8.70905 
8.71  156 

8.7i  395 
8.71  63§ 

8.70714 
8.70962 
8.71  203 

8.71  453 
8.71  697 

1.29286 
1.29038 
1.2879! 
1.28546 
1.28  303 

9-99943 
9-99943 
9-99942 
9.99942 
9.99941 

5 
4 
3 

2 

I 

60 

8.71  880 

8.71  939 

1.28060 

9.99940 

0 

Log.  Cos. 

D 

Log.  Cot. 

Com.  D. 

Log.  Tan. 

Log.  Sin. 

/ 

350 


TABLE  VII.  —  LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 


1 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

p.  p.                  1 

0 

I 

2 

3 

4 

8.71  880 
8.72  120 
8.72359 

8.72  597 
8.72833 

246 
239 
237 
236 
235 
233 
233 
231 
230 
229 
227 

226 
22§ 
224 
223 
221 
221 
2I§ 
2I§ 

8.71  939 
8.72  186 
8.72426 
8.72659 
8.72896 

241 
240 
238- 
237 
235 
235 
233 
232 

231 
229 

22§ 
22? 
226 
225 
22§ 
223 
221 
220 
219 

218 

217 
216 

214 
214 
213 

212 
210 
210 
209 

207 

207 
206 
204 
204 
203 
202 
2OI 
200 
199 
I98 

197 
197 
195 
195 
194 

193 
192 
*| 

196 
190 
183 
«8§ 
187 
i8g 
185 
185 
184 
183 
182 
182 

1.28066 
1.27819 

1.27  579 

1.27341 

1.27  104 

9.99940 
9.99940 
9.99939 
9-99938 
9.99938 

60 

59 
58 

I 

6 

7 
8 
9 

10 
20 

30 
40 
50 

6 

I 

9 

10 
20 

30 
40 

5° 
6 

1 

9 

10 

20 
30 
40 
50 

6 
9 

10 

20 

30 
40 
50 

6 
7 
8 
9 

10 
20 

3° 
40 
50 

6 

1 

9 

10 

20 

30 
40 

5° 

330 

33-0 
38.5 
44-o 
49-5 
55-o 

IIO.O 

165.0 
220.  o 

275.0 

290 

29.0 

1 

145.0 

193.3 
241-6 

250, 

25.0 
29.1 

33-3 
37-5 

&i 

125.0 

i66.£ 
208.3 

2X0 

21.0 
24-5 
28.0 

31-5 

35-o 
70.0 
105.0 
140.0 
175-0 

9. 

0.9    o 
I.I    I 

320 

32-0 
37-3 
42-6 
48.0 

53-3 
106.6 
160.0 
2'3-3 
266-6 

280 

28.0 
32-6 
37-3 
42.0 

46-6 
93-3 
140.0 
i86.§ 
233-3 

240 

24.0 
28.0 
32.0 
36.0 
40.0 
80.0 

120.0 

160.0 

200.0 

200 

20.0 

2l'§ 
26.6 
30.0 

8:1 

100.  0 

!33-3 
166.6 

9     8 

.9   0.8 
.6   0.9 

310 

31.0 
36-  i 
4i-3 
46-5 
5i-6 
103.3 
i55-o 
206.6 

258-3 

270 

27.0 
3i-5 
36.0 
40-5 
45-o 
90.0 
i35-o 
180.0 
225.0 

230 

23.0 
26.3 
30-6 

I1- 

76-5 
115.0 

153-3 
191.6 

190 

19.0 

22.1 
25-3 
28-5 

I'-e 

63-3 

95-0 
126.6 
158.3 

7     < 

0.7    o 
0.8    o 

300 

30.0 

35-0 
40.0 
45-o 
50.0 

IOO.O 

150.0 

200.0 

250.0 

260 

26.0 
3°-3 

34-6 
39-° 
43-3 
86-6 
130.0 
!73-3 
216-6 

22O 

22.  O 

ii 

m 

73-3 

IIO.O 

&i 

180 

18.0 

21.0 

24.0 
27.0 
30.0 

60.0 
90.0 

120.0 
.150.0 

6    0.5 
7    0.6 

8 
9 

8.73069 
8.73302 
8-73  535 
8-73  766 
8.73997 

8.7313? 
8.73366 
8.73  599 
8.73831 
8.74062 

1.26863 
1.26633 

1  .  26  406 
1.26  163 
1.2593? 

9-99937 
9-99936 
9-99935 
9-99935 
9-99934 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 
H 

8.74226 

8-74453 
8.74680 

8.74905 
8.75  129 

8.74292 
8.74526 
8.74748 
8-74974 
8.75  199 

1.25708 
1.25479 
1.25252 

1.25026 
1.24801 

9-99933 
9-99933 
9-99  932 
9-99931 
9-99931 

50 

49 

48 

47 
46 

II 

17 

18 
19 

8-75353 
8.75  574 
8-75795 
8.76015 
8.76233 

8.75422 

8.75645 
8.75867 
8.76087 
8.7630^ 

1.24577 

1.24354 
1.24133 
1.23913 
1.23693 

9.99930 
9.99929 
9.99923 
9.99928 
9-99927 

45 
44 
43 
42 
4i 

20 

21 
22 
23 
24 

8.76451 
8.7666? 
8.76883 
8.77097 
8.77310 

217 
2lg 

«l 

214 
213 

212 
2IX 

210 
209 
208 
207 
206 
205 

204 

203 

202 
201 
200 

199 

'98 
197 
197 
193 
J95 
194 
i93 
192 
191 
191 
189 
189 
188 
187 
iSg 

iflj 

185 
184 
183 

182 
182 

181 

8.76524: 
8.7674? 
8.76958 
8.77  172 
8.77  38§ 

1.23475 
1.23253 

1.23042 

1.22  82? 
I.226I3 

9.99925 

9-99925 
9.99925 

9-99924 
9.99923 

40 

P 

% 

3 

27 
28 

29 

8-77  522 
8.77733 
8.77943 
8.78  152 
8.78366 

8-77  599 
8.77811 
8.78022 
8.78232 
8.78441 

1.22  400 
1.22  l8§ 
1.  21  978 
1.  21  768 

i.  21  559 

9.99922 
9.99  922 
9.99921 
9.99926 
9.99919 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 

34 

8.78  567 
8.78773 
8.78973 
8.79183 
8.79386 

8.78643 
8.78855 
8.79061 
8.79266 
8.79470 

1.  21  35! 
1.  21  144: 
1.20938 
1.20734 
1.20  530 

9.99919 
9.99918 
9.9991? 
9.99  915 
9.99  916 

30 

29 

28 
27 
26 

1 

13* 
39 

8.79588 
8.79789 

8.79989 
8.80  189 
8.80387 
"8.80585 
8.80782 
8.8097? 
8.81  172 
8.  8  1  365 

8.79673 

8.79875 
8.80075 
8.80275 
8.80476 

1.20327 
1.20  125 
I.I9923 
I.I9723 
I.I9524 

9.99915 
9.99914 

9.999I3 
9.99912 
9.99912 

25 

24 

23 

22 
21 

40 

4i 

42 

43 
44 

8.80674 
8.80871 
8.81  063 
8.  8  1  264 
8.81  459 

I.I9326 
I.I9  I2§ 
1.18931 
1.18736 

1.18  541 

9.99911 
9-99  9io 
9-99  909 
9.99  903 

9-99  90? 

20 

19 
18 

17 
16 

1.4  I 

1.6    i 
3-5    3 

*:!  i 

7-9    7 

4     ' 

0.4    o 
0.5    o 

-3    1-2 
•5    i-3 

.0     2.6 

•5    4-o 
•°    |-3 
•5    6-6 

I       3 

1    °-2 
4    0.3 

i  .0 
i.i 

2-3 

3-5 
4-0 
5-8 

2 

0.2 
O.2 

o 

I 

2 
3 

4 
5 

; 

0. 

o. 

?    °-7 

o    o.  o 

o    1-6 
o    2.5 
o    3-2 

0     4.f 

[       d 

i    o.o 
i    0.6 
i    o.o 

I      O.I 
I      O.I 

3    o.i 
5    0.2 

$  0.3 

8    0-4 

45 
46 
47 
48 

49 

8.81  560 
8.81  752 
8.  8  1  943 
8.82  134 
8.82324 

8.81  653 
8.  8  1  846 
8.82033 
8.82230 
8.82426 

1.18347 
1.18  154 
1.17961 
I.I7770 
1.17579 

9-99907 
9.99906 

9.99905 
9-99  904 
9.99903 

15 
H 
13 

12 
II 

50 

5i 
52 
53 

54 

8.82513 
8.82  701 
8.82883 
8.83075 
8.83266 

8.82616 
8.82799 
8.8298? 
8.83175 
8.83361 

1.17389 

I.I7  201 
I.I70I2 
I.I6825 
1.16633 

9.99902 
9.99902 
9.99901 
9.99900 
9.99899 

10 

9 
8 

6 

°'Z    ° 
0.7    o 

'•5   ' 

2.2      2 

3-0      2 

3-7    3 

6    0.4 

6  °-s 

3    x.o 
o    1.5 

§    2'° 
3    2.5 

°-2 
°'3 
o.6 
I.O 

••i 

*-6 

0. 

o. 

0. 
0. 

o 

0. 

P 
P 

59 

8.83445 
8.83629 
8.83813 
8.83995 
8.8417? 

8.83  547 
8.83732 
8.83915 
8.84  loo 
8.84282 

1.16453 
I.I6268 
I.l6o83 
I.I5900 
I.I57I? 

9.99893 
9.9989? 
9.99895 
9.99896 
9.99895 

5 
4 
3 

2 

60 

8.84358 

8.84464 

I-I5535 

9.99894 

0 

Log.  Cos.        d. 

Log.  Cot. 

c.  d.  .    Log.  Tan. 

Log.  Sin. 

/ 

P.  P. 

S6< 


351 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS, 


r 

Los,'.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

P.  p 

0 

2 

3 

4 

8.84358 
8.84538 
8.84718 
8.84897 
8.85075 

1  80 
180 

i?8 
178 

8.84464 
8.84645 
8.84826 
8.85005 
8.85  184 

181 
1  85 

'79 
179 

I.I5535 
I-I5354 
I.I5  174 
I.I4994 
I.I48I5 

9.99894 
9.99893 
9.99892 
9.9989! 
9.99896 

60 

59 
58 

I 

6 

181 

18.1 

21.  I 

1  80 

18.0 

21.0 

178 

I7.8 
20.7 

176 

17.6 
20.5 

5 

8.85252 

177 
17? 

8.85363 

J78 

1.14637 

9.99  889 

55 

8 

24-1 

24.0 

23-7 

23-4 

6 

7 
8 

9 

8.85429 
8.85605 
8.85780 
8.85954 

176 
J75 
J74 

8.85  546 
8.85717 
8.85893 
8.86063 

177 

176 
176 

I7§ 

I.I4459 
1.14283 
I.I4  107 
LI393I 

9.99883 
9.99888 
9.99887 
9.99886 

54 
53 
52 
5i 

10 

20 

3° 
40 

50 

1°-; 
60.3 

90.5 

120.5 

150.  §  1 

30.0 
60.0 
90.0 

120.0 

150.0 

29.6 

59-3 
89.0 
ii8.fi 
148.3 

20.4 

2Q-3 
58.6 
88.0    i 

l\l'l 

10 

ii 

12 
13 
14 

8.86  123 
8.86301 
8.86474 
8.86645 
8.86815 

174 
173 
172 
171 
171 

8.86243 
8.8641? 
8.86596 
8.86763 
8.86935 

J75 
174 
173 

172 
172 

1.  13756 
1.13582 
1.13409 
1.13237 
1.13065 

9.99885 
9.99884 
9.99883 
9.99882 
9.9988! 

50 

49 
48 

47 
46 

6 

7 
g 

174 

17.4 
20.3 

172 

17.2 
20.  o 

170 

17.0 
iQ-8 

168 

16.8 
19.6 

15 

16 

17 
18 

19 

8.86987 
8.87155 
8.87  325 
8.87494 
8.87661 

170 
169 
169 
i6§ 
167 

8.87  105 
8.87277 

8.87447 
8.87616 

8.87785 

171 
170 
170 

169 
169 

1.12893 
1.12723 

1.12553 
1.12384 

1.  12  215 

9.99886 
9.99879 
9-99878 
9.9987? 
9.99875 

45 
44 
43 
42 
41 

9 
10 
20 
30 

40 
50 

26.  1 
29.0 

58.0 

87.0 
116.0 
145.0 

22.9 

25.8 

28.6 

57-3 
86.0 
114.5 
143-3 

25-5 

ai:2 

56.6 
85.0 

us-? 
141-6 

25.2 

28.0   ; 
56.0 
84.0 

112.  0 
I4O.O 

20 

21 

22 

23 
I      24 

8.87823 

8.87995 
8.88  166 
8.88326 
8.88496 

167 
165 
165 
,  i«5 
164 

8.87953 

8.88  120 
8.88287 
8.88453 
8.88613 

167 
167 
166 

165 

I.I2047 
1.  1  1  880 
I.U7I3 

1.  1  1  547 

1.1138! 

9.99875 
9.99874 
9.99874 

9.99873 
9.99872 

40 

39 

38 

% 

6 

166 

16.6 
i9-3 

22.  I 

164 

16.4 
19.1 

21.  § 

162 

16.2 
18.9 

21.6 

160 

16.0 

l8'S    ' 
21.3    i 

25 
26 

27 
28 

29 

8.88654 
8.88  81? 
8.88980 
8.89  142 
8.89303 

163 
163 
162 
162 
161 

8.88783 
8.8894? 
8.89111 
8.89274 
8.8943g 

165 
164 
163 
'63 
162 

i.  u  215 
i.  ii  052 
1.10889 
1.10726 
1.10563 

9.99871 

9-99  870 
9.99869 
9.99868 
9.99867 

35 
34 
33 
32 
31 

9 

10 

20 
30 
40 

5° 

24.9 
27-6 

55-3 
83.0 
110.5 
138-3 

24.6 
27-3 

54-6 
82.0 
109.3 

136-6 

24-3 

27.0 

54-° 
81.0 
108.0 

i35-o 

24.0 

11  : 

80.0 
io6-$  ' 

'33-3 

30 

3i 
32 
33 
34 

8.89464 
8.89624 
8.89784 
8.89943 
8.90  101 

160 
159 
*59 
158 

__0 

8.89593 
8.89759 
8.89920 
8.90080 
8.90  240 

161 
161 
160 
159 

1.1040! 
1.10246 
1.10079 
1.09919 
1.09760 

9.99866 
9.99865 
9.99864 
9.99  863 
9.99862 

30 

29 
28 
27 
26 

6 

7 

8 

^ 

s:| 

21.0 

156 

\l:l 

20.8 

154 

15-4 
17.9 
20.5 

152 

15-2 
17.7    : 

2O.  2 
22    8 

P 

37 
38 
39 

8.90  259 
8.90417 

8.90  573 
8.90729 
8.90885 

150 
157 

156 
156 
156 

8-90398 
8.90557 
8.90714 
8.90872 
8.91  023 

J58 
158 
J57 
IS7 
156 

Tcfi 

1.0960! 
1.09443 
1.09285 
1.09  128 
1.0897! 

9.99  861 
9.99860 
9.99859 
9.99858 
9.99857 

25 
24 

23 

22 
21 

10 

20 
30 
40 
50 

26.3 

52-6 
79.0 

Jos-s 
131-6 

26.0 

52.0 

78.0 
104.0 
130.0 

25-6 

5J-3 
77.0 
102.5 
i28.§ 

25-3 
50-6      | 
76.0 

IO'-3    ! 
126.5 

40 

41 

42 
43 
44 

8.91  046 
8.91  195 
8.91  349 
8.91  502 
8.91655 

154 
154 
153 
153 

jcS 

8.91  184 
8.91  340 
8.91  495 
8.91  649 
8.91  803 

I5° 
155 
155 
154 
154 

Itr5 

1.08815 
1.08660 
i.  08  505 
1.08356 
i.  08  iQg 

9.99856 
9'99855 
9.99853 
9.99852 
9.9985! 

20 

19 

18 

17 
16 

6 
7 
8 
9 

150 

15.0 

r7-5 

20.0 
22.5 

149 

14.9 
17.4 

'9-§ 
22.3 

148 

14.8 
17.2 
19.7 
22.  '.; 

147 

14.7 

17.  i 
19.6 

22.0 

45 
46 

47 
48 

49 

8.91  807 
8.91  959 
8.92  1  16 
8.92  261 
8.92411 

151 

•5? 

150 
150 

8.91  957 
8.92  109 
8.92  262 
8.92413 
8.92565 

152 

J52 

151 
«ss 

I  eo 

1.08043 
1.07  896 
1.07738 
1.07  5^6 
1-07435 

9.99856 
9.99849 
9.99843 
9.99847 
9.99845 

15 

14 
13 

12 
II 

20 
30 

5° 

25.0 
50.0 

75-° 

IOO.O 

125.0 

24-8 
49-6 
74-5 
99-3 
124.1 

24-6 
49-3 
74  o 

98  6 
123-3 

24-5 

49-0 
73-5 
98.0 
122.5 

50 

5i 

52 
53 
54 

8.92  561 
8.92710 
8.92853 
8.93007 
8.93  154 

149 

M8 
148 
147 

8.92  715 
8.92866 
8.93015 
8.93  164 
8-933IS 

150 
149 
149 
149 

T  .0 

1.07284 
1.07  134 
1.06984 
1.06835 
i.o668§ 

9.99845 
9.99844 
9.99843 
9.99842 
9.99841 

10 

9 

8 

6 

6 

I 

9 

10 

146 

14.6 
17.0 

19  4 
21.9 
24.3 

145 

14-5 
16.9 

I9.| 

21.7 
24.1 

I 

O.  I 
0.2       ( 
O    2 
O.2       < 
0.2       < 

i     6 

3.1      O.O 
3.1      O.O 
3.1      O.O 
3.1      O.I 
3.1      O.I 

55 
56 

P 

59 

8.9330! 
8.93448 

8-93  594 
8.93  740 
8.93  885 

J47 
148 
146 
I46 

145 

8.9346? 
8.93609 
8  93  756 
8.93903 
8.94049 

148 

*47 

M6 
146 

i.  06  533 
1.06396 
1.06243 
1.06097 
1.05956 

9.99840 
9.99839 
9-99837 
9-99836 
9-99835 

5 
4 
3 

2 
I 

20 
30 
40 
5° 

48.6 
73-o 
97-| 

121.  g 

48.3 
72-5 
96.5 

120.  § 

0  § 
0.7     < 

I.O       < 
1.2 

3.3    o.i 

3-5      0.2 
3-6      °'§ 

3-8    0.4 

60 

8.94029 

144 

8-94  195 

*45 

1.05  805 

9.99834 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

t 

P.    1 

1 

85 


352 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

5° 


Log.  Sin.        d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

E 

.    P. 

0 

I 

2 

3 
4 

8.94029 
8.94174 

8-9431? 
8.94466 
8.94603 

144 
143 

J43 
142 
142 
141 
141 
140 
140 
i39 
139 
138 
J38 
138 
137 
137 
136 
136 

8-94  195 
8.94346 
8.94485 
8.94629 
8-94773 

145 

144 
144  ^ 

1  44 

142 
142 
142 
141 
141 
141 
140 
140 
'39 
139 
i3§ 
138 
137 

137 
136 
136 
135 
»35 
134 
134 
133 
133 
133 

131 

130 
130 
129 
129 
129 

128 
127 
127 

^26 

126 
125 
125 

125 

124 
124 
124 
124 
123 
123 
123 

122 
122 
121 

1.05  805 
1.05659 
1.05515 
1.05  376 
1.05  225 

9.99834 

9.99832 
9.99831 
9.99830 

60 

59 
58 
57 
56 

6 

\ 

9 

10 
20 

JO 

50 

6 

\ 

<* 
to 

2   : 

SB 

6 
7 

3 
9 

10 
20 

30 

40 

6 

I 

Q 

10 

•o 

33 

4  J 

9* 

6 

~ 

: 
.' 

: 
•2 

:- 
+  • 
50 

•' 

7 

.- 

9 

i 
2  :• 

y* 

145 

14-5 
16.9 
19-3 
21.7 

ii 

120-8 

140 

14.0 

£| 

21.0 

46.6 
70.0 

.3:1 

13! 

iS. 

20. 
22. 

00. 
112. 

133 

13- 

15- 

19- 
21. 

43- 
65- 
87. 
109. 

127 

12.7 
14-8 
16.9 
19.6 

21  .1 
42-3 
63.5 

i?J:i 

122 

12.2 
14.2 

16.2 

18.3 
20.3 
40.6 

61.0 
81.3 
101.5 

I 
i 

i 
i 

2 

2 

7 
9 

12 

I 

I 
1 
I 

2 
2 

i 

9 

i  : 

5 
7 

0 
2 

5 

0 

5 
o 
5 

t 

i 

§ 

5 

i 

X 
X 

x 

I 

2 

4 

• 

: 

I  . 

I 

I 
I 

I 
2 
4 

1 

i 

10 

*4 

4-4 

3.3 

M 

1.6 

S:S 

2.0 

5.0 

D.O 

39 

11 
»J 

;  •  o 

3 

5-8 

i: 

i 

2 
2 

4 

6 
I 

ii 

*: 
i 

a 

4 

6 
1 

tc 

26 

2.'' 

8:2 

I.O 

3-« 

4- 

5-° 

21 

2  . 
4. 

6. 

5. 

0, 
>.  ; 
>.  = 
t>.| 

o.  § 

S:| 

19.0 

21.4 

23-8 

47-6 
7x-5 
95-3 
119.1 

3 

16.1 
18.4 
20.7 

si 

69.0 

92.0 

115.0 

*4    J 

3-4 
5-6 
7-8 

3.1 

'-i 

t-6 
7-o 

?-i 

t-6    z 
JO      1 

j.O 

5-5 
7-3 
?•§ 

3-3 
5.0 

5  -6 

i-3     i 

125 

12.5 
14.6 
16.6 
18.7 

20.§ 
41-6 
62.  S 

83-| 
104.1 

120 

12.0 
14.0 

16.0 
18.0 
20.  o 
40.0 
60.0 
80.0 

100.  0 

i4! 

16.  t 

i8.c 

2I-3 
23-6 
47-3 
71.0 

94-^ 
118.3 

137 

13-7 
16.0 
18.2 

20.  i 

22-g 

45-6 
68.  = 

114.1 

33 

13-3 

17.7 
19.3 

22.1 

44-3 
66.5 

88'6 
10-8 

29 

12.9 
15.0 
17.2 
'9-3 
21.5 
43  o 
64-5 
86.0 

07-5 
124 

III 

16.5 

18.5 
20-  1 

f-3 
62.0 

82.^ 
103.3 

i 

O.I  0 
O.2  O 
0.2  0 
0.20 
0.20 
0.50 

0.7  o 
i.olo 
i.  3|o 

141 

14  i 
16.4 
18.8 

21.  I 
23-5 

47-o 
70.5 
94-o 
II7-5 

,36 

13.6 

.1:1 
20.4 

22.^ 

45-3 
68.0 
9o.fi 
"3-3 

132 

13.2 
15.4 
17.6 
19.8 

22.0 

44.0 

56.o 
88.0 

IIO.O 

128 

12.8 

149 
17.0 
19.2 
21.3 

42-6 
64.0 

85-3 
106.6 

123 

12.3 
14-3 
16.4 
18.4 
20.5 
41.0 
oi  .5 
82.0 
102.5 

i    6 

.1  O.O 

.  I   O.O    ' 

.10.6 

.1  O.I 
.1  O.I 

.30.1 

lo'f 

I 

7 
8 

9 

8.94745 
8.94887 
8.95023 
8.95  169 
8.95310 

8.94917 
8.95059 

8.95  202 

8-95  344 
8.95485 

1.05083 
1.04946 
1.04798 

1.04656 

1.04514 

9.99829 
9.9982? 
9.99826 
9.99825 
9.99824 

55 
54 
53 

52 

10 

ii 

12 

13 

14 

8.95  450 
8-95  58§ 
8.95  728 
8.95  867 
8.96005 

8.95625 
8.9576? 
8-95  90? 
8.9604? 
8.96  i8g 

1.04373 
1.04232 

1.04092 

1.03952 
1.03813 

9.99823 
9.99822 
9.99821 

9.998l§ 

50 

49 
48 
47 

46 

17 

18 
19 

8.96  143 
8.96280 
8.96417 
8.96553 
8.96689 

8.96325 
8.96464 
8.96602 

8.96739 
8-96  876 

1.03674 

1.03  536 
1.03  398 

1.03  266 

1.03  123 

9.9981? 
9.99815 

9.99815 
9.99814 
9.99813 

45 
44 
43 
42 

20 

21 

22 
23 
24 

8.96825 
8.96960 
8.97094 
8.97229 
8.9?  363 

135 
134 
134 
134 
133 
J33 
132 
132 
132 
131 

130 

130 

13° 
129 

129 

I2§ 

127 

127 

I26 
126 
126 
125 
125 
125 
124 
124 
124 
123 
123 

122 

122 
122 
122 
121 
121 
I2O 
120 

8.97013 

8-97  149 
8.97285 
8.97421 
8.97  556 

1.02985 

1.  02  856 
1.  02  714 
1.02  579 

1.02444 

9.99811 
9.99816 
9.99  809 
9.99808 
9.99807 

40 

39 
38 

25 
26 
27 
28 
29 

8.97496 
8.97629 
8.97  762 
8.97894 
8.98026 

8.97696 
8.97  825 

8-97958 
8.98092 
8.98225 

1.02  309 
1.02  175 
I.0204I 

i.oi  908 
i.oi  775 

9.99805 
9.99804 
9.99803 
9.99802 
9.99801 

35 
34 
33 
32 

30 

32 
33 
1    34 

8.08  m? 

8.98  283 

8.98419 

8.98  549 
8.98  679 

8.9835? 
8.98490 
8.9862? 

8.98  753 
8.98  884 

i.oi  642 
i.oi  510 
i.oi  378 
i.oi  247 

i.oi  116 

9-99799 
9-99798 
9-99797 
9-99796 
9-99794 

30 

29 

28 

27 
26 

P 

37 
38 

39 

8.98  8o§ 
8.9893? 
8.99066 

8-99  194 
8.99322 

8.99015 
8.99  145 
8.99275 
8.99404 
8-99  533 

1.00985 
1.00855 
1.00725 
1.00595 
1.00466 

9.99793 
9-99  792 
9-99  79i 
9-99789 

25 
24 
23 

22 
21 

40 

42 
43 
44 

8.99449 
8.99  577 
8.99703 
8.99830 
8.99956 

8.99662 
8.99791 
8.99919 
9.00045 
9.00  174 

1.00337 
1.00209 
1.00081 

0-99953 
0.99826 

9.99787 
9.99786 
9-99784 
9.99783 
9.99782 

20 

19 
18 

16 

45 
46 
47 
48 
49 

9.00081 
9.00  207 
9-oo  332 
9.00455 
9.00  586 

9.00300 
9.00427 
9.00553 
9.00679 
9.00  804 

0.99699 

0-99  573 
0.99445 
0.99321 
0.99  195 

9.99781 
9-99779 
9-99778 
9-99777 
9-99776 

15 
13 

12 
II 

50 

52 
53 

54 

9.00704 
9.00828 
9.00951 
9.01  073 
9.01  196 

9.00930 
9.01  054 
9.01  179 
9-oi  303 
9.01  427 

0.99  070 
0.98945 
0.98821 
0.98697 
0.98  573 

9-99774 
9-99773 
9-99772 
9.99776 

9-99769 

10 

6 

:  I 

59 

9.01  318 
9.01  440 
9.01  561 
9.01  682 
9.01  803 

9.01  550 
9-oi  673 
9.01  796 
9.01  913 
9.02  046 

0.98450 
0.98327 
0.98  204 
0.98081 
0-97959 

9.99768 
9-99766 
9.99765 
9-99764 
9.99763 

5 

4 
3 

2 
I 

60 

9.01  923 

9.02  162 

0-97  838 

9.99761 

0 

Log.  Cos.        d. 

Log.  Cot.   i   c.  d.  1    Log.  Tan. 

Log.  Sin. 

' 

p.  P.                 j 

84' 


353 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

6° 


r 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

p.  p. 

0 

I 
2 

3 

4 

9.01  923 
9.02  043 
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30 

29 
28 
27 
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25 
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9.99675 

0 

Log.  Cos.        d. 

Log.  Cot.       c.  d.      Log.  Tan.     Log.  Sin.         ' 

P.P. 

83' 


354 


TABLE  VII. -LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

7° 


1 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

P.  P. 

0 

I 

2 

3 

4 

9.08  589 
9.08  692 
9.08  794 
9.08  897 
9.08  999 

102 
102 
I  O2 
102 

9.08914 
9.09013 
9.09123 
9.09  226 
9.09330 

104 

104 
^ 
103 

103 

T    5 

0.91  085 
0.90981 
0.90877 
0.90773 
0.90670 

9.99675 
9.99673 
9.99672 
9.99676 
9.99669 

60 

P 
1 

6 

7 
8 

9 

9.09  lor 

9.09  202 
9.09  303 
9.09  404 
9.09  505 

101 
101 
101 
101 

9-09433 

9-09  536 
9.09  639 
9.09  742 
9.09844 

103 
103 
103 

102 
102 

0.90  566 
0.90463 
0.90  366 
0.90  258 
0.00155 

9.9966? 
9.99665 
9.99664 
9.99662 
9.99661 

55 
54 
53 
52 
5i 

6 
7 
8 
9 

10 

20 

104 

10.4 

12    I 
13-8 

15  6 
l7-3 
34-6 

103 

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12.0 

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11.9 
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n.  8 
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12 
13 
1       H 

9.09606 
9.09  706 
9.09805 
909900" 
9.  IOOO6 

IOO 
IOO 
100 

99 

9.09  947 
9-IOQ48 
9.10  156 
9.10252 
9-10353 

101 
102 
101 
101 

0.90053 
0.8995? 
0.89849 
0.89748 
0.89647 

9.99659 
9.99658 
9.99656 
9.99654 
9-99653 

50 

49 
48 

47 
46 

3° 
40 

5° 

69-3 
86-6 

&-I 
85-8 

68.0 
85.0 

H 

15 

16 

17 
18 

19 

9.10  105 
9.  10  205 
9.  10  303 
9.10402 
9.10  501 

99 
99 
98 
99 
98 

9.10454 
9-10555 
9.10655 
9.10  756 
9.10  856 

101 
101 
IOO 
IOO 
IOO 

0.89  546 
0.89445 
0.89  344 
0.89  244 
0.89  144 

9.9965! 
9.99650 
9.99648 

9.99646 
9.99645 

45 
44 
43 
42 
4i 

1 

7 
8 
9 
10 

IOO 

lo.o 
11.7 
13-4 
15-1 
16.7 

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99 

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20 

21 
22 
23 
24 

9.10  599 
9.10697 

9-  I0  795 
9.10892 
9.10990 

93 
98 
97 
97 
97 

9.10  956 
9.11055 
9.11  155 
9  ii  254: 
9-  1  1  353 

99 
99 
99 
99 

o.  89  044 
0.88  944 
0.88845 
0.88745 
0.88645 

9.99643 
9.99641 
9.99640 

9-99638 
9.99637 

40 

39 
38 
37 
36 

20 
30 
40 
50 

33-5 

5: 

83-7 

33-3 
50.0 

66-6 
83-3 

33-o 
49-5 
66.0 
82.5 

32-6  ! 
49.0 

65-3 

Si.g 

3 

27 
28 
29 

9.11087 
9.11  184 
9.11  281 
9-  "37? 

9-  i  i  473 

97 
96 
97 
96 
96 

9.11  452 
9.11  556 
9.  1  1  649 

9-  1  1  747 
9.  1  1  845 

98 
98 
98 
98 
98 

0.88  548 
0.88  449 
0.88  351 
0.88  253 
0.88  155 

9.99635 
9-99633 
9.99632 

9-99  630 
9.99623 

35 

34 
33 
32 
3i 

6 

9? 

9-7 
11.4 
13.0 

97 

9-7 
11.3 
12.9 

96 

9-6 

II  .2 
12.8 

95 

9-5 
ii.  i 

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30 

3i 
32 
33 
34 

9.11  570 
9.11665 
9.11761 

9-11856 
9.11  952 

96 
95 
96 

95 
95 

9-  i  1  943 

9.  1  2  046 

9-12  13? 

9.12235 
9.1233! 

98 
97 
97 
97 
96 

0.88  057 
0.87959 
0.87  862 
0.87  765 
0.87663 

9.99627 
9.99625 

9-99623 
9.99622 
9.99  620 

80 

29 
28 
27 
26 

9 
10 

20 

3° 
40 
50 

14.6 
16.2 

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65.0 
81.2 

J4-5 
16.1 
32-3 
48-5 
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14-4 

16.0^ 
32.0 
48.0 
64.0 
80.0 

14.2 

15-3 
3i-6 
«.7  -5 
63-2   -, 
79-1 

35 
36 

37 
38 
39 

9.12047 
9.12  141 
9.12236 
9.12336 
9.12425 

95 
94 
94 
94 
94 

9.12428 
9.12525 
9-12  621 
9.I27I7 

9.12  813 

97 
96 
96 
96 
96 

0.87  57i 
0.87475 
0.87  379 
0.87  283 
0.87  187 

9.99613 
9.99617 
9.99615 
9.99613 
9.99611 

25 
24 

23 

22 
21 

6 

94 

9-4 

94 

9-4 

93 

9-3 

92 

9-* 

40 

4i 
42 

43 
44 

9.12513 
9.12  612 
9.12  706 
9.12799 

9.  I  2  892 

93 
94 
93 
93 
93 

9.12903 
9.13004 
9.13099 

9-!3  194 
9.13289 

95 
95 
95 
95 
95 

0.87  091 
0.86996 
0.86  906 
0.86805 
0.86716 

9.99610 
9.99603 
9.99606 
9.99605 
9.99603 

20 

I  Q 

17 

16 

I 

9 
10 
20 

30 
40 

II.  0 
12.6 

14.2 
15-7 

3J-5 
47.2 
63.0 
78.7 

12-5 

14.1 

i5-£ 
31-3 
47-o 
62.  A 

78.3 

10.  g 

12.4 
13.9 
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31.0 

46-5 
62.0 

77-5 

12.2 

13-8 
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3°-6 
46.0 

51 

76-5  • 

4J 

8 

49 

9.12985 
9.13078 
9.13  170 
9.13263 
9-13355 

93 
92 
92 
92 
92 

9.13384 
9-13478 
9.13572 
9-I366S 
9.13766 

94 
94 
94 
94 
94 

n5 

0.86616 
0.86  521 
0.8642? 
0.86  333 
0.86239 

9.99601 
9.99600 
9.99  598 
9-99  596 
9-99  594 

15 

H 
13 

12 
II 

QI 

01 

00      2 

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50 

5i 

52 
53 
54 

9-13447 

9  13  538 
9.13636 
9.13721 
9-13813 

9? 
92 
9i 

9; 
~ 

9.13854 

9-^394? 
9.14041 

9-  HI  34 
9.14227 

93 
93 
93 
93 
93 

0.86  146 
0.86052 
0.85959 
0.85  866 
0.85773 

9-99  593 
9-99  59i 
9-99  589 
9-99  58? 
9-99  586 

10 

6 

6 
7 
8 
9 

10 
20 

3° 

9.1 
10.7 

12.2 

J3-Z 
15.2 

30.5 
45-7 

9.1      « 
10.6     K 

12.  1       I 

'3-6     * 

15.1    I 

30-3     3< 
45-5     4 

J.O       0. 

3.5     o. 
z.o     o. 
5-5     o. 
;.o    o. 

5.0      0. 

;.o     i. 

2       O.I 
2       O.2 
2       O.2 

2     o-2. 
3     0.2 

6     o-5 
o     0.7 

55 
56 

% 

59 

9-i39°3 
9-13994 
9.14085 

9-H  175 
9.14265 

9° 
9i 
90 
95 
90 

9.14319 
9.14412 
9.14504 

9-  H  596 
9.14688 

92 
92 
92 
92 
92 

0.85  686 
0.85  588 
0.85495 
0.85  403 
0.85311 

9.99  584 
9-99  582 
9.99  586 

9-99579 
9-99577 

5 

4 
3 

2 

I 

40 
50 

61  .0 
76.2 

6o.g     & 
75-8     7 

3.O       I  . 

5.0      I. 

3     '-° 
6     i-a 

60 

9-H35S 

90 

9.14786 

92 

0.85  219 

9-99  575 

0 

Log.  Cos. 

d. 

Log.  Cot. 

C.  (1. 

Log.  Tan. 

Log.  Sin. 

r 

P.  P. 

82 


355 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS 

8° 


/ 

Log.  Sin     '     d. 

Log.  Tan. 

c.  d. 

M>g.  Cot. 

Log.  Cos. 

p.  p. 

0 

2 

3 
4 

Q 

8 
9 

9-  H355 
9.14445 

9-H535 
9.14624 

9-I47I3 

90 
89 
89 
89 
89 
89 
88 
88 
88 
88 
88 
88 
8? 
87 
8? 
87 
87 
8£ 
8£ 
88 
86 
86 
86 
85 
85 
85 

85 

85 

84 

84 
84 
84 

84 
84 

84 
83 
83 
83 
83 

0, 

9.14786 
9.14872 
9.14963 
9.15054 
9.15  145 

9? 
91 

9* 

9i 

9i 
96 
96 
90 
90 

89 
90 
89 
89 
89 

89 
89 

89 
88 
88 

88 

oo 
oo 

8? 
88 

8? 

87 
8? 

87 
86 
87 

86 
86 

86 
86 

85 

86 

85 
85 

85 
85 

85 
84 
84 
84 
84 
84 
84 
84 
83 
83 

83 
83 

83 
83 

82 

82 
82 
82 
82 
82 
82 

0.85  219 
0.85  128 
0.85037 
0.84945 
0.84854 

9-99575 
9-99  <73 
9.99  371 

9-99  570 
9-99  568 

60 

59 

I 

6 

8 
9 

10 
20 
30 
40 
50 

6 

8 
9 

10 

20 

30 
40 

50 

6 

8 
9 

10 
20 
30 
40 
50 

6 

8 
9 

10 
20 
30 

40 

50 

9* 

9.i 
10.7 

12.2 

13-7 
I5.2 

30.5 

45-f 
61.0 
76.2 

88 

8-8 
10.3 
11.8 
13-3 

14-? 
29-5 
44.2 
59.0 
73-? 

85 

8.5 

10.0 

11.4 

12.8 

14.2 
28.5 
42.? 
57.o 
71.2 

82 

8.2 

9-6 

II.  0 

12.4 

13.? 

27.5 
41,2 

m 

6 

8    i 
9    i 

10     I 
2O     2 

30   3 
40    5 
50  6 

9i 

9.1 
10.6 

I2.T 

13-6 
15.! 

30-3 
45-5 
60.5 
75-8 

88 

8.8 

10.2 

II.? 
13.2 
14-6 
29.3 
44.0 

58.5 

73-3 

85 

8.5 
9-9 
n-3 

12.? 

14.1 
28.3 
42.5 
56-6 
70-8 

82 

8.2 

9.5 

10.9 

12.3 

13-6 

27.3 

41.0 

54=6 
68.3 

79 

7.9   o 
9-3   o 
0.6   o 
1.9   o 
3-2   o 
6.5    o 

9.?  I 

3.0    i 

6.2     I 

90 

9.0 

10.5 

12.0 

!3-5 

15.0 
30.0 
45.0 

60.0 

75.0 

87 

8.7 

10.  1 

ii.  6 

13-5 
14.5 
29.0 

43-5 
58.0 

72-5 

84 

8.4 
9-8 

II.  2 
12.6 
14.0 
28.0 
42.0 
56.0 
70.0 

81 
8.1 

9-4 
10.8 

12,! 

!3-5 
27.0 
40.5 
54-0 
67.5 

2       i 

.2    0. 
.2     0. 
.2     0. 
•3     0- 
•3     0- 

-6   o. 

.0     0. 

.§  I. 

•  6    i. 

89 

8.9 

IO./ 

ii.  8 

13.3 
14-8 
29.6 

44-5 
59-3 
74-  f 

86 

8.6 
lo.o 
11.4 
12.9 

14-3 
28.5 
43-o 
57-3| 
71-6 

83 

8.3 

9-7 
ii.  6 
12.4 
13-  § 
27-6 
41.5 
55.3 
69.1 

80 

8.0 

9-3 

IQ.| 

12.0 

13-3 
26.6 
40.0 

III 

1 
2 
2 

2 
2 

D 

2 

9.  14  802 
9.14891 
9.14980 
9.15068 
9-I5I57 

9.15236 
9.15327 

9.I54I? 
9.1550? 
9.15598 

0.84763 
0.84673 
0.84582 
0.84492 
0.84402 

9.99  565 
9-99  564 
9-99563 
9.99  561 

9-99559 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 
14 

9.15245 

9-15333 
9.15421 

9-15508 
9-15595 

9.1568? 
9.1577? 
9.15867 

9-15956 
9.16045 

0.84312 
0.84222 
0.84  133 
o.  84  043 

0.83954 

9-9955? 
9-99555 
9-99  553 
9-99552 
9-99550 

50 

49 
48 

47 
46 

IS 

16 

17 
18 

19 

9.15683 
9.15770 
9.I5857 

9-15943 
9.  16030 

9.16134 
9.16  223 
9.16312 
9.16401 
9.16489 

0.83865 
0.83775 
0.8368? 

0.83  599 
0.83  511 

9.99548 
9-99  546 
9-99  544 
9-99  542 
9.99541 

45 

44 
43 
42 

4i 

!    20 

21 
22 

23 

1      24 

9.16115 

9.  1  6  202 

9.l628§ 
9.16374 
9.16460 

9  1657? 
9.16665 
9.16753 
9.16841 
9.16928 

0.83  422 
0.83334 
0.83  247 
0.83  159 
0.83  071 

9-99539 
9-99  537 
9-99  535 
9-99  533 
9-99  53T 

40 

39 
38 

g 

25 
26 

27 
28 
29 

9.16545 
9.16636 
9.16716 
9.I680I 
9.16885 

9.17015 
9.17  103 
9.17  190 
9.17275 
9-I7363 

0.82984 
0.82  897 
0.82  810 
0.82  723 
0.82635 

9.99529 
9.99528 
9-99  526 
9-99  524 
9.99522 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 

34 

9.16970 
9.17054 
9.I7I39 
9.17223 
9.17307 

9.1745° 
9.17536 
9.17622 
9.17703 
9  17794 

0.82  550 
0.82  464 
0.8237? 
0.82  291 
0.82  206 

9.99526 
9.99518 
9-99516 
9-995H 
9.99512 

30 

29 
28 

27 
26 

35 
36 

37 
38 
39 

9.I739I 

9  17474 
9.17558 
9.17641 
9.17724 

9  17880 
9.17965 
9.18051 
9.18  136 

9.  1  8  221 

0.82  1  20 
o.  82  034 
0.8  1  949 
0.8  1  864 
0.81  779 

9.99511 

9-99  5°9 
9.99507 

9-99  505 
9-99  503 

25 
24 
23 

22 
21 

40 

4i 

42 

43 
44 

9.17807 
9.17890 
9.17972 
9.18055 
9-1813? 

°3 
83 
82 
82 
82 
82 
82 
82 
81 
81 
8! 
81 
81 
86 
81 

9.18306 
9.18396 
9.18475 
9.18559 

9.  1  8  644 

0.81  694 
0.8  1  609 
0.81  525 
0.8  1  446 
0.81  356 

9-99  501 
9-99499 
9.9949? 
9.99495 
9-99493 

20 

19 
18 

17 
16 

!  4I 
46 

47 
48 

49 

9.18219 
9.18301 
9-18383 
9.18465 
9.18545 

9.18728 
9.18812 
9.18896 
9.18979 
9.19063 

0.81  272 
0.81  1  88 
0.81  104 
0.81  026 
0.80937 

9.99491 
9.99489 
9.9948? 

9.99485 
9.99484 

15 

H 
13 

12 
II 

50 

5i 
52 
53 

i     54 

9.18628 
9.18709 
9.18790 
9.18871 
9.18952 

9.19  146 
9.19229 
9.19312 

9-19395 
9.19478 

0.80854 
0.80776 
0.8068? 
o.  80  604 
0.80  522 

9.99482 
9.99480 
9.99478 
9.99476 
9-99474 

10 

1 

7 
6 

i  p 

57 
58 
59 

9.19032 
9.19113 
9.19193 
9  19273 
9.19353 

oo 
86 
86 
80 
80 

79 

9.19  566 
9.19643 
9.19725 
9.1980? 
9.19889 
9.19971 

0.80439 
0.80357 
0.80274 
0.80  192 
0.80  1  16 

9.99472 
9.99470 
9.99468 
9.99466 
9-99464 

5 
4 
3 

2 

60 

9-19433 

0.80  02§ 

9.99462 

0 

Log.  Cot.       c.  d.      Log.  Tan.      Log.  Sin. 

f 

P.P. 

O  -I  o 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

9° 


/ 

Log.  Sin.   1     d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

p. 

p. 

0 

2 

3 

4 

9-19433 

9-I95I3 
9.19592 
9.19672 
9.19751 

80 

79 
79 
79 
79 
79 
79 
78 
78 
78 
78 
78 
78 
/8 
7? 
7? 
71 
77 
77 
77 
77 
76 
76 
76 
76 
76 
76 

a 

7$ 
75 
75 
7$ 

» 

74 

75 
74 
74 
74 
74 
74 
74 
73 
74 
73 
73 
73 
73 
73 
73 
72 
73 
72 
72 

72 
72 
72 
72 
72 

71 

9.19971 
9.20053 
9.20  134 
9.20216 
9.  2O  297 

81 
81 

8i 
81 
81 
Si 
81 
86 
8l 
So 
86 
80 
80 
80 

79 
79 
79 
79 
79 
79 
79 
78 
78 
78 
78 
78 
78 
78 
78 
7? 
7? 
7? 
7f 
77 
77 
77 
76 
77 
76 
76 

76 
76 
76 

76 
7$ 
7$ 
75 
75 
75 
75 
75 
75 
74 
74 
74 
74 
74 
74 
74 

0.80  02  8 

0.79947 

0.79865 

0.79784 
0.79703 

9.99462 
9.99460 
9.99458 
9.99456 

9-99454 

00 

3 

i 

6 

8 
9 

10 
20 

30 
40 
50 

t 

6 

8 
9 

10 

20 
30 
40 
50 

. 

4 

6 

8 
9 

10 
20 

30 
40 
50 

8 

8 
9 

10 
12 
13 

27 

40 

54 
67 

6 

8 

9 

0 
10 

>0 

^o 

>0 
7 

1C 

1  1 
i: 

2- 

^ 
5' 

6: 

6 

8 

9 

[0 
20 

30 

*o 

$0 

I 

I 

2 

3 
4 
5 

[ 
I 

.2 

.6 
.1 
-7 

•9 

7 
7 

9 

10 

ii 

13 

26 

39 

5- 
65 

6 

•6 
-9 

).2 
J 

.0 

•7 

7, 

8 

9 
1  1 

12 

24 

:/ 

49 

61 

7i 

% 

9-5 

D.7 

i-9 

3-8 
5-7 

7-(, 
9.6 

8 
8 

9 

10 
12 
13 
^7 

40 
54 
67 

8 

•4 
.8 
.  i 
.? 

.2 

•  3 
-4 

> 

/ 

1C 

ii 

i: 

2( 

3* 
5< 

6: 

3 

i 

.8 
.0 
.2 
•  5 

•  7 
.0 

.2 

t 

1 
I 
2 

3 
4 
5< 

i      8< 
.1     8 

•4     9 
.8  10 

.1     12 

-5  13 
.0  26 

-5  4o 
•o  53 
.5  66 

78 

7-8 
9.1 
10.4 
11.7 
13.0 
26.0 

39-o 
52.0 
65.0 

6      7 

.6     7 
.8     8 

>.I     10 

.4  ii 
-6  12 
-3  25 
>-o  37 
>.g  50 

1-3:62 

73 

1:1 

9-1 
10.9 

12.  1 

24-3 
36.5 
48-6 
6o.g 

7i      * 
7.1    o 
5.3    o 
M   o 
3.6   o 
i.  §   o 
3-6   o 
5-5    i 
7-3    i 

?.T     2 

3 
O 

I 

0 

0 

3 

6 

i 

I< 

I 
I 
2 

3 

6 

5 

.0 
.2 

•  5 

.0 

•5 

.0 

•5 

» 

i 
i 

2 

3 
4 

6 

> 

2 

3 
3 

4 
4 
8 

6 

i 

79 

7-9 
9.2 
10.5 
ii.  8 
13.1 
26.3 
39-5 
52-6 
65-8 

n 

7-7 
?.o 

D.2 

1.5 

2-8 

if 

1.3 

I-I 

74 

7-4 
8.6 

98 
i  i.i 

12.3 
24-6 
37-o 
49-3 
6i-6 

72 

7-2 

8.4 
9-6 

3.8 

2.0 
4-0 

5.o 
8.0 

D.O 

2 
0.2 
0.2 
O.2 

o-3 
o-3 
0-6 

1.0 

i-3 
1.6 

I 

7 
8 

9 

9.19830 
9.19909 
9.19988 
9.20065 
9.20145 

9.20378 
9.20459 
9.  20  540 
9.  20  620 
9.  20  701 

0.79622 

0.79541 
0.79460 
0.79379 
0.79298 

9.99452 
9.99450 
9.99448 
9.99446 
9-99444 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 
14 

9.20223 

9.  20  301 
9.20379 
9.2045? 

9.20535 

9.2078! 
9.20862 
9.  20  942 
9-21  022 
9.21   102 

0.79218 

0.79138 
0.79058 
0.78978 

0.78898 

0.78818 
0.78739 
0.78  659 
0.78  580 

0.78  501 

9-99442 
9.99440 

9.99!43? 
9-9943$ 
9-99433 

50 

49 
48 
47 
46 

15 

1  6 

17 
18 

19 

9.20613 

9.  20  696 
9.  20  768 
9.20845 
9.2O  922 

9.21  181 

9.21  26l 

9-2i  346 

9-21  420 

9-2i  499 

9-99431 
9.99429 
9.9942? 
9-9942$ 
9-99423 

45 

44 
43 
42 

4i 

20 

21 
22 

23 
24 

9.20Q99 
9.21  076 
9.21  152 
9-21  229 

9.21  305 

9.21  578 
9.21  657 
9-2i  73$ 
9.21  814 
9.21  892 

0.78422 
0.78  343 
0.78  264 
0.78  1  86 
0.78  10? 

9.99421 
9.99419 
9.99417 
9.99415 
9-994I3 

40 

38 
37 
36 

11 

27 
28 
29 

9.21  382 
9.21  458 
9.21  534 
9.21  609 
9.21  68$ 

9.21  971 
9.22049 
9.22  127 
9.22  205 
9.22  283 

0.78  029 

0-77  95i 
0.77  873 
0.77  795 
0-77  717 

9.99411 

9-99408 
9.99406 
9.99404 
9.99402 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 
34 

9.21  761 

9-21  836 
9.21  911 

9.21987 

9.22062 

9.22366 
9.22438 
9.22.515 
9-22  593 
9.22  670 

0.77  639 
0.77  562 
0.77  484 
0.77  407 
0.77  330 

9.99406 
9-99398 
9-99  396 
9-99394 
9.99392 

30 

29 
28 
27 
26 

11 
11 

39 

9-22  135 
9.22  2iT 
9.22  286 
9.22  366 

9.22435 

9.22747 
9.22824 

9-22  000 
9.2297? 
9.23054 

0.77  253 
0.77  176 
0.77099 
0.77022 
0.76946 

9-99389 
9.9938? 
9-9938$ 
9-99383 
9.99381 

25 
24 
23 

22 
21 

40 

4i 
42 
43 
i    44 

9.22509 
9.22583 
9.2265? 
9.22731 
9.22  805 

9.23130 
9.23206 
9.23  282 

9-23358 
9  23  434 

0.76  870 
0.76793 
0.7671? 
0.76641 
0.76  56§ 

9-99379 
9-99377 
9-99374 
9-99372 
9.99376 

20 

17 
16 

45 
46 
47 
48 
49 

9-22878 
9.22952 
9.23025 

9.23093 
9.23  171 

9.23516 
9.23586 
9.23661 

9-23737 
9.23812 

0.76489 
0.76414 

0.76338 
0.76  263 
0.76  1  88 

9.99368 
9.99366 
9.99364 
9.99361 
9-99359 

15 
14 
13 

12 
II 

50 

5i 

52 
53 
54 

9.23244 

9-233I7 
9.23390 
9.23462 
9-23535 

9-23887 
9.23962 
9.24037 

9.  24  1  1  2 
9.24l8§ 

0.76  113 
0.76038 
0.75963 
0.75888 
0.75813 

9-99357 
9-99355 
9-99353 
9-9935° 
9-99348 

10 

I 

7 
6 

55 
56 

.  H 

59 

9.2360? 
9.23679 
9.23751 
9.23823 
9.23895 

9.24  26l 

9-2433$ 
9.24409 
9.24484 
9-24558 

0.75739 
0.75664 
0.75  596 
0.75516 
0.75442 

9-99346 
9-99344 
9-99342 
9-99  339 
9-9933? 

5 

4 
3 

2 
I 

60 

9.23967 

9.24632 

0-75  368 

9-99335 

0 

Log.  C'08.         d. 

Log.  Cot.   i   c.  d. 

Log.  Tan. 

Log.  Sin. 

* 

P 

P. 

8OC 


357 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  , COTANGENTS. 


/ 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

p.  p.                j 

0 

I 

2 

3 

'      4 

9.23967 

9.24  038 
9.24  no 
9.24  181 
9.24252 

71 
7t 
7i 
7i 
7i 
7i 
7i 
7i 
76 

76 
76 
7o 
76 
70 

69 
70 

69 
70 

69 
69 
69 
69 
69 

69 
68 
69 
68 
68 
68 

68 
68 
68 
68 
6? 
6? 
68 
67 
6? 
6? 
67 
67 
67 
66 
67 
66 
66 
66 
66 

66 

66 
66 

65 

66 
65 
65 
65 

8 

65 

65 

9.24632 
9.24705 
9.24779 
9.24853 

9.24926 

73 
74 
73 
73 
73 
73 
73 
73 
73 

73 
72 
72 
72 
72 

72 

72 

72 
72 
7i 
72 
71 
7i 
7* 
7i 
7* 

% 

7i 
76 

76 
76 
76 
70 
76 
70 
70 

69 
70 

69 
69 

§ 

6? 

69 
69 

69 
69 
68 
69 
68 
68 
68 

68 
68 

6? 
68 
6? 
6? 
6? 

0.75  368 
0.75  294 
0.75  220 
0.75  147 
0.75073 

9-99335 
9.99333 
9-99  336 
9-99328 
9-99326 

tiO 

11 
1 

6 

8 
9 

10 

20 
30 
40 
50 

6 

8 
9 

10 
20 
30 
40 
50 

6 

8 
9 

10 

20 

30 
40 
50 

6 

8 

9 
10 

20 
30 
40 

50 

7 
6     7 
7      8 
8     g 

9   ii 

10     12 

20    24 

30   37 
40  49 
50   61 

72 
7.2 
8.4 
9-6 
10.9 

12.  1 

24.1 
36.2 
48.3 

60.4 

76 

7.0 

8.2 

9-4 
10.6 

ii.? 

23-5 
35-2 
47-o 
58-? 
68 

6-8 
8.0 

9-  i 
10.3 
11.4 

22.§ 
34-2 

45-6 
57-1 
66 
6.6 
7.? 
8.8 

10.0 

ii.  i 

22.T 

33-2 

44-3 
55-4 

6 

b 
9 

1C 

20 

30 
40 

50 

4       ' 

-4    ; 
•6    •* 

8     < 
.1    i 

•3    i- 

•6     2/ 

.0   3< 
•3   4< 

.6  6 

72 

7-2 

8.4 
9.6 
10.8 

12.0 
24.0 
36.0 
48.0 
60.0 

70 

7-o 
8.1 

9-3 
10.5 

ii-6 
23.3 
35-o 
46-6 
58-3 
68 
6.8 

7-9 
9.6 

10.2 

ii.  3 

22.6 
34-0 

45-3 
56-6 
66 
6.6 

7.7 
8.8 

9-9 

II.  0 

22.0 

33-0 
44-o 
55.0 

2 

O.2 

o-3 
o-3 
0.4 

0.4 
o.§ 

'   1.2 

i-6 

2.1 

73 

'•3      ' 
5.6 

>.8      < 

.0     I< 

>.§    i: 
1-5    2, 
>•?    3< 
).o   4< 

.2     6( 

72 

7-i 
8-3 
9.5 
10.7 
11.9 
23-8 
35-? 
47-6 
59-6 

69 

I:? 

9.2 

10.4 
1  1.6 

23-1 
34-? 
46.3 
57-9 
6? 

6.? 

7.9 
9.0 

10.  I 
II.  2 

22.5 

33'? 
45.0 
56.2 

6$ 

6-5 
7.6 

8.? 
9-8 
10.9 

2I.§ 

32-? 
43-6 
54-6 

2 

0.2 
0.2 
0.2 
°-3 
°-3 

0.6 

I.O 

1*6 

73 

7-3 

3.5 
?.? 
:>.§ 

2.1 
*-j 

5*6 
*l 

71 

7.1 
8.3 
9.4! 
10.6 

ii.  8 

23-6 

35-5 
47-3 
59-i 
69 

6.9! 
8.6 
9.2 
10.3 

n-5 
23.0 

34-5 
46.0 

57-5 
67 

6-7 
7-8 
8.9 
lo.o 
ii.  T 
22.3 
33-5 
44-6 
55-8 
65  i 
6.5! 
7-6 

8.6 

9-? 
10.8 

21.6 

III 

54.  i 

6 

8 
9 

9.24323 
9.24394 
9.24465 

9-24536 
9.24607 

9.25  ooo 

9.25073 
9.25  146 
9.25219 

9:25  292 

0.75  ooo 

0.74927 
0.74854 
0.74781 

o.  74  708 

9.99324 
9.99321 
9-993I9 
9.99317 
9-993I5 

55 

54 
53 
52 
5i 

10 

ii 

12 
13 

14 

9.2467? 
9.24  748 
9.24818 
9.24883 
9-24958 

9.25  365 

9-2543? 
9.25  510 
9.25  582 
9.25654 

0.74635 
0.74  562 
0.74490 
0.7441? 
0-7434? 

9.99312 
9.99316 
9.99308 
9.99306 
9-99303 

50 

49 
48 

47 
46 

11 

17 

18 
19 

9.25028 
9.25098 
9.25  i6f 
9-2523? 
925306 

9-25  727 
9.25799 
9.25871 

9.25  943 
9.26  014 

0.74273 

0.74  201 

0.74129 
0.74057 
0.7398^ 

9.99301 
9.99299 
9.99296 
9-99294 
9.99292 

45 
44 
43 
42 

4i 

20 

21 
22 
23 
24 

9.25376 
9.25445 
9.25  514 

9-25583 
9.25652 

9.26086 
9.26  158 
9.26  229 
9.26  306 
9.2637! 

0.73913 
0.73  842 
0.73  771 
0.73  699 
0.73623 

9.99290 
9-9928? 
9.99285 
9.99283 
9.99  286 

40 

39 
38 

1 

25 
26 

27 
28 

29 

9.25721 
9.25790 
9.25853 
9.25927 
9.25995 

9-26443 
9.26514 
9.26584 
9.26655 
9.26  726 

0-73557 
0.73  486 
0.73415 
0-73344 
0.73274 

9-99278 
9.99276 
9.99273 
9.99271 
9.99269 

35 
•34 
33 
32 
3i 

30 

3i 
32 
33 
1    34 

9.  26  063 
9.26  131 
9.26  199 
9.26  26? 
9-26335 

9.26  79g 
9.26867 

9-2693? 
9.2700; 
9.27078 

0.73203 

o.73  133 
0.73  062 
0.72  992 
0.72  922 

9-99  26g 
9.99264 
9.99  262 

9.99259 
9.99257 

30 

29 
28 
27 
26 

35 
36 

% 

39 

9.  26  402 
9.26476 
9-2653? 

9.  20  605 

9.26672 

9.27  148 
9.27218 
9.27  28? 

9-27  35? 
9.27427 

0.72  852 
0.72  782 
0.72712 
0.72  642 
0.72  573 

9.99255 
9.99252 
9.99250' 
9-99  248 
9.99245 

25 
24 
23 

22 
21 

40 

4i 

42 

43 
44 

9.26739 
9.2680§ 
9-26873 
9.  26  940 
9.27007 

9-27496 
9.27  566 
9.27635 
9.27704 
9.27773 

0.72  503 
0.72434 
0.72  365 
0.72  295 
0.72  226 

9.99243 
9.99  246 
9.99238 
9.99236 
9-99233 

20 

19 

18 

17 
16 

45 
46 

47 
48 

49 

9.27073 
9.27  140 
9.27206 
9.27272 
9-27339 

9.27842 
9.27911 
9.27  980 
9.  28  049 
9.28  ii? 

0.72  15? 
0.72  083 
0.72020 

0.71  951 
0.71  882 

9.99231 
9-99228 
9-99226 
9.99224 
9.99221 

15 
14 
13 

12 
II 

50 

5i 
52 
53 

54 

9.27405 
9.27471 

9-27  536 
9.  27  602 
9.27668 

9.28  1  86 
9.28254 
9.28  322 
9-28396 
9.28459 

0,71  814 
0.71  746 
0.71  67? 
0.71  609 
0.71  541 

9.99219 

9-99216 
9.99214 
9.99212 
9.99209 

10 

9 
8 

6 

% 
% 

59 

9.27733 
9.27799 
9.2786^ 

9.27929 
9.27995 

9.28  527 
9.28  594 
9.28662 
9.28730 
9.2879? 

0.71  473 
0.71  405 

0.71  33? 
0.71  270 

0.71  202 

9.99207 
9.99204 

9.99  202 

9-99  199 
9-99  197 

5 
4 
3 

2 

I 

60 

9.  28  060 

9.28865 

0.71   135 

9  99  194 

0 

Log.  Cos.  I      d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

/ 

P.  P. 

358 


TA.BLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

11° 


t 

Lew.  sin.         «1. 

Loe.  Tan.      r.  d.       LOST.  Cot. 

LOST.  Cos. 

p. 

P. 

0 

2 

3 

4 

9.28  060 
9.28125 
9.28  189 
9.28254 
9.28319 

65 
64 
65 
64 
64 
64 
64 
64 
64 
64 
64 
63 
64 
63 
63 
63 
63 
63 
63 
63 
63 
63 
62 

63 
62 
62 
62 
62 
62 
62 
62 
62 
62 
61 
62 
61 
61 
61 
61 
61 
61 
61 
61 
61 
65 
61 
66 
66 
66 
66 
60 
60 
66 
59 
60 
60 

59 
60 

59 

59 

9.28  865 
9.28932 
9.29000 
9.29067 
9.29134 

6? 
67, 
67 
67 
67 
6£ 
67 
6§ 
67 
66 

66 
66 

'    66 
66 
66 
66 
65 
65 

65 

6 

63 

65 

^ 

65 

65 
65 
64 

i\ 

64 

64 
64 

64 

64 
64 
64 
64 
63 
64 
64 
63 

% 

63 

63 
63 
63 
63 
63 
63 

6^ 

62 

63 

62 
62 
62 
62 
62 
62 

0.71  135 

0.71  067 
0.71  ooo 
0.70933 
0.70866 

9-99  194 

9-99  192 
9-99  l89 
9.99  187 
9.99  185 

00 

p 
i 

6 

8 

9 
10 

20 

30 
40 
50 

6 

7 
8 

9 

10 
20 

30 
40 

5° 

6 

7 
8 

9 
10 

20 

30 
40 

5° 

6 

( 
7 

1C 

1  1 

22 

32 
4J 

55 

6 

( 

i 

9 

1C 
21 

5- 
43 
55 
6 
6 

8 

9 

!<: 

20 

31 
41 

5- 
6 

8 

9 
10 

20 
30 

;o 
50 

I 

j 

'< 
•( 

( 

K 
2C 

3< 
4< 

5< 

6 

8 
9 

0 

:o 

JO 
*o 

5^ 
6 

.i, 

•7 

•8 

.0 

!i 

.2 
•3 

•4 
4 

4 

:I 

•  r 
.7 

:I 

.0 

•7 

2 
.§ 

•5 

4 
•4 
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t 

1 

( 
I< 

2. 

5< 
4< 
5< 

S 

7 

3 

J 
> 

3 
} 

) 
) 

61 
6. 
7- 
9- 

10 

ii 

22 

33 
45 
56 
6 
6 

8 

9 
ii 

22 

33 
44 
55 
6, 
6 

8 
9 

10 
21 

32 

42 

53 
6: 

6. 

S. 
9- 

10. 
20. 

31- 

41. 

51- 
>6 

5.6 

7.6 

3.6 

?-I 
XI 
XI 

X2 

>-3 
X4 

3 

0-3 
0.3 
0.4 

0.4 

0.5 

I.O 

i-5 

2.0 

2.5 

r 

7 

9 

0 

I 

2 

5 

0 
2 

5 

r, 

•7 
8 

•9 

0 
.0 
0 
.0 

0 

1 

4 

4 

6 

0 

8 

5 
I 

2 
2 
2 

I 

6 

0 

3 
§ 
t 
(. 

< 

K 
2C 

y 

4< 

s< 

0 
0 
0 
0 
0 
0 

I 

2 

67 

6.7 
7-8 

8-9 
lo.o 
ii.  i 
22.3 
33-5 
44-6 
55-8 
6$ 
6.5 

7-6 
8.? 

9-8 
10.9 

21-8 

32.? 
43-6 
54-6 

63 

6-3 
7-4 
8.4 
9-5 
10.6 

21.  I 

31.? 
42.3 

52.9 

61 

6.1 

7.2 

8.2 

9.2 

10.2 
20.5 

30-? 
41.0 
51.2 
)0 

5.0 

7.0 
5.0 

>o 

xo 

XO     I 
XO     2 

xo   3 
xo   4 

2         2 

.2     0. 

•3   o. 
3   o. 
4  o. 
4  o. 
8   o.( 

2     I.< 

6    i. 
i    i.( 

65 

6-5 
7.6 

8-6 
io.§ 

21-6 

32.5 
43.3 
54.1 
63 
6.3 

8^4 
9-4 
10.5 

21.0 

31-5 
42.0 

52-5 

61 

6.1 

7-i 
8.T 

9-t 

10.  I 

20.3 
30-5 
40-6 
50-8 
59 

5-9 
6.9 
7-9 
8.9 
9-9 
9-8 
9-? 
9-6 
9.6 

2 
2 
2 

3 

3 

6 

8 
9 

9-28383 
9.28448 
9.28512 
9.28575 
9.28641 

9.29  20! 
9.  29  268 

9-29335 
9.29401 

9.29468 

0.70793 
0.70732 
0.70  665 
0.70  598 
0.70531 

9.99  182 
9.99  1  80 
9.99  17? 
9.99175 
9.99  172 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 
U 

9.28705 
9.28769 
9.28832 
9.28895 
9.  28  960 

9-29535 
9.29601 
9.29667 

9-29734 
9.29800 

0.70465 
0.70393 
0.70  332 
o.  70  266 

0.  70  200 

9.99  170 
9-99  l67 
9-99  165 
9.99  162 
9.99  1  60 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

9.29023 
9.29087 
9.29  150 
9.29213 
9.29277 

9.  29  866 
9.29932 
9.29998 
9.30064 
9-30*29 
9.30  195 
9.  30  266 
9.30326 
9.3039I 
9-30456 

0.70  134 

o.  70  068 

0.  70  002 
0.69  936 
0.69876 

9.99157 
9.99155 
9.99152 
9.99150 
9.99  147 

45 
44 
43 
42 

4i 

20 

21 
22 

23 

24 

9.29340 
9.29403 
9.  29  466 

9.29528 
9.2959! 

0.69  805 
0.69739 
0.69  674 

0.69  6o§ 
0.69  543 

9.99  145 
9.99  142 
9.99I39 
9.99  137 
9-99  134 

40 

39 
38 

1 

3 

27 
28 
29 

9.29654 
9.29715 
9.29779 
929841 
9.29903 

9.30522 

9.30587 
9.30652 

9.30717 
9.30781 

0.69478 
0.69  413 
'0.69  348 
0.69  283 
0.69  2i§ 

9.99  132 
9-99  129 
9.99127 
9.99  124 
9.99  122 

35 

34 
33 
32 
3i 

30 

3i 
32 
33 
34 

9.29965 
9.3002? 
9.  30  089 

9-30*5* 

9.30213 

9.30845 
9.30911 

9.30975 
9.31040 
9.31  104 

0.69  153 
0.69089 
0.69024 
0.68  960 
0.68  896 

9.99119 
9.99115 
9.99114 
9.99111 
9-99  109 

30 

29 
28 
27 
26 

P 

37 
38 
39 

9-33275 

9-  3°  336 
9-30398 
9  3°  459 
9.30526 

9.31  i6§ 
9.31  232 
9.31  297 
9.31  361 
9.31  424 

0.68  831 
0.68  76? 
0.68  703 
0.68  639 
0.68  575 

9-99  105 
9-99  I04 
9.99  101 
9.99098 
9.99096 

25 
24 
23 

22 
21 

40 

41 
42 
43 
44 

9.30582 

9-30643 
9.30704 
9.30765 
9.30  826 

9-3I488 
9-3i  552 
9.31  616 
9.31  679 
9-3i  743 

0.68  511 
0.68  44? 
0.68  384 
0.68  326 
0.68  257 

9.99093 
9.99091 
9.99088 
9.99085 
9.99083 

20 

19 

18 

17 
16 

4! 

47 
48 

49 

9.30885 

9  30  947 
9.31  008 
9.31  o6§ 
9.31  129 

9.31  805 
9.31  869 

9-3i  933 
9.31  996 
9-32059 

0.68  193 
0.68  136 
0.68067 
0.68  004 
0.67  941 

9.99  086 

9.9907? 
9.99075 
9.99072 
9.99069 

15 
14 
13 

12 
II 

50 

51 
52 
53 
54 

9.31  189 
9.31  249 
9-31  309 
9-31  370 
9.31429 

9.32  122 
9-32  185 
9.32  248 
9.32316 
9-32  373 

0.67  878 
0.67  815 
0.67  752 
0.67  689 
0.67  625 

9.99067 
9.99064 
9.99062 
9.99059 
9.99055 

10 

1 

7 
6 

% 

57 
58 
59 

9.31489 

9.3i  549 
9.31609 
9.31669 
9-31  728 

9-32436 

9-32498 
9.32  566 
9.32623 
9.32685 

9o274? 

0.67  564 
0.67  501 
0.67  439 
0.67  377 
0.67  314 
0.67  252 

9.99054 
9.99051 

9.99048 
9.99046 

9-99043 

5 

4 
3 

2 
I 

60 

9.31  7& 

9.99040 

0 

Log.  Cos.   i     d. 

Log.  Cot.    i   c.  d.       Log.  Tan. 

Log.  Sin. 

' 

p. 

I'. 

78C 


359 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 


Log.  Siu. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

p. 

p. 

0 

I 

2 

3 

i      4 

9-31  788 
9-31  84? 

9-31  9°6 
9.31  066 
9.32025 

59 
59 
59 
59 
59 
59 
59 
58 
59 
58 
58 
58 
58 
58 

58 
58 
58 
58 
58 
58 

I 

5? 
5? 
57 
5? 
57 
57 
57 
57 
57 

5 

56 
56 

11 

56 
56 

*  56 
55 
56 

55 
55 
55 
55 
55 
55 
55 
55 

a 

55 
55 

9-32747 
9.32809 
9-3287? 
9-32933 
9-32995 

62 
62 
62 
62 

61 
61 

62 
61 
61 
61 
61 
61 
61 
61 
66 
61 
61 
66 
61 
66 
65 
66 
60 
66 
60 
66 
60 
60 
59 
60 
60 
59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
59 
58 
58 
59 

58 
58 
58 
58 
58 
58 
58 
5? 

9 

58 

0.67  252 
0.67  196 
0.67  I2§ 
o.  67  065 
0.67004 

9.99040 
9.99038 

9-99035 
9-99032 
9.99029 

60 

P 
g 

6 

7 
8 

9 

10 
20 
30 
40 
50 

6 

8 
9 

10 
20 
30 

40 
50 

6 

8 
9 

10 

20 
30 
40 
50 

t 

6     t 

*7         * 

9     < 

10     I( 
2O     2( 

3°   3 

40   4 

50   5 

66 

6.6 
7.6 
8.6 
9.1 

10.  1 

20.  T 

30.2 

40.3 
50.4 

5l 

6.  8 

7-8 

80 
.0 

9-? 
19.5 
29.2 

39-o 
48.? 

56 

5-6 
6.6 

7-5 
8-5 
9-4 
i8.§ 
28.2 

37-6 

47-1 

6 

8 
9 

10 

20 
30   : 

40  : 

50   < 

2         t 

>.2        t 

7.2        > 

5.2      1 
)-3      < 

X§     I( 
D-5     2< 

[.o    3< 

i-3    4 
1-6    5 

60 

6.0 
7-0 
8.0 
9.0 

10.0 

20.  o 
30.0 
40.0 
50.0 

58 

5-8 
6.? 

7-? 
8.7 
9-6 
19-3 
29.0 

38-6 
48.3 

56 

5-6 
6.5 

74 
8.4 
9-3 
18.6 
28.0 

37-3 
46.5 

54 

5-4   c 

6.3   < 

7.2    c 

8.2     ( 

9.1    c 
8.1 

>7.2 

56.3  : 

L5-4    : 

>i 

>.!      < 

7.2 
5.2 
}-2 
X2     K 
D-5     2( 
5-?     3 

[.o   4 
i.2    5 

59 

5-9 
6.9 

7-9 
8.9 

9-9 

19-8 
29.? 

39-6 
49-6 

5? 

a 

7.6 

8.6 
9.6 
19.1 

28.? 
38.3 
47-9 

5S 

8 

7-4 

9-i 

18.5 

27.? 

37-0 
46.2 

3     ; 

3-3    o 
3.3    o 
).4   o 
5.4   o 
3.5    o 

[.O    O 

.5  I 

5.0     I 
J.  t\     2 

61 

7-1 
3.1 

M 
11 

3-3 
3.5 
3.5 

D.§ 

59 

I'9 

6.9 

7-8 
8-8 
9-8 
19-6 
29.5 

39-3 
49-1 

57 

11 

7-6 
8-5 
9-5 
19.0 
28.5 
38.0 
47-5 

55 

7^3 

8.21 

9.1 
18.3 
27-5! 
36.6 
45-8 

\ 

2 

4 
4 
I 

2 

6 
i 

8 
9 

9.32084 

9-32  143 
9.32202 
9.32  266 
9-323I9 

9-33057 

9-33  H8 
9-33  1  86 

9.33303 

0.66  943 
0.66  88T 
0.66  819 
0.66758 
0.66695 

9.99027 
9.99024 
9.99021 
9.99019 
9.99016 

55 
54 
53 
52 

10 

ii 

12 
13 
14 

9.32378 
9-32436 
9.32495 
9-32  553 
9.32611 

9.33364 

9-33487 
9-33548 
9.33609 

0.66635 
0.66  574 
0.66  513 
0.66452 
0.66  396 

9.99013 
9.99016 
9.99008 
9.99005 
9-99  OO2 

50 

49 
48 

47 
46 

11 

17 

18 
19 

9.32670 
9.32728 
9-32786 
9.32844 
9.32902 

9-33670 
9-33731 
9-33792 
9-33852 
9.339I3 

0.66330 
0.66  269 
0.66  208 
0.66  14? 
0.66085 

9.98999 

9.98  997 
9.98  994 
9.98991 
9.98  98§ 

45 

44 
43 

42 

20 

21 

22 

23 
24 

9.32  960 
9-33oi? 
9-33075 
9-33  133 
9-33  196 

9-33974 
9-34034 
9-34095 
9.34155 
9-342I5 

9-34275 
9-34336 
9-34396 
9-34456 
9-345I5 

0.66026 
0.65  965 
0.65  905 
0.65  845 
0.65  784 

9.98986 

9-98983 
9.98  986 
9.9897? 
9.98975 

40 

39 
38 
37 
36 

27 
28 
29 

9.33  248 
9-33305 
9-33  362 
9-334I9 
9-33476 

0.65  724 
0.65  664 
0.65  604 
0.65  544 
0.65  484 

9.98972 

9.98969 
9.98965 
9.98963 
9.98  961 

35 

34 
33 
32 
31 

30 

32 
33 

1    34 

9-33533 
9-33  590 

9.33704 

9-34575 

9-34695 
9-34754 
9-348i4 

0.65  424 
0.65  364 
0.65  305 
0.65  245 
0.65  1  86 

9-98958 

9.98952 
9.98949 
9.98947 

30 

29 
28 
27 
26 

1 

!     39 

9.33874 
933930 

9-34043 

9-34873 
9-34933 
9-34992 
9.35051 
9.35  116 

0.65  125 
0.65  067 
0.65  008 
0.64943 
0.64  889 

9.98944 
9.98  941 

25 
24 

23 

22 
21 

i    40 

42 
43 
44 

9.34099 
9.34156 
9.34212 
9.  34  268 
9-34324 

9.35  169 
9-35228 
9.3528? 

9-35346 
9-35405 

0.64  836 
0.64771 
0.64712 
0.64653 
0.64  594 

9.98930 
9.98927 

9.98921 
9.98913 

20 

19 
18 

17 
16 

46 
47 
48 
49 

9-34379 
9-34435 
9-34491 
9-34547 
9.34602 

9-35464 
9-35  522 
9.35581 
9.35640 
9.35698 

0.64  536 
0.6447? 
0.64418 
0.64360 
o.  64  302 

9.98915 
9.98913 
9.98  910 
9.98907 
9.98  904 

15 
14 
13 

12 
II 

i    50 

52 
53 
54 

9.34658 
9-347I3 
9-34768 
9.34824 

9-34879 
9-34934 

9-35044 
9-35099 
9-35  J54 

9-35756 
9-35815 
9.35873 
9.35931 
9-35989 

0.64  243 
0.64  185 
0.64  127 
0.64063 
0.64016 

9.9890! 

9;  98  895 
9.98  892 
9.98  890 

10 

9 
8 

6 

i 

59 

9.3604? 
9.36105 
9.36163 
9.36221 
9.36273 

0.63952 
0.63894 
0.63  837 
0.63  779 
0.63  721 
0.63663 

9.98887 
9.98884 
9.98881 
9.98878 
9.98875 

5 
4 
3 

2 
I 

60 

9-35209 

9-36338 

9.98  872 

0 

Log.  Cos.         d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

' 

p.  P.                   i 

afio 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

13° 


t 

Log.  si,,. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

P. 

p. 

0 

I 

2 

3 
4 

9-35  209 
9-35  263 
9-353I8 
9-35  372 
9-35427 

54 
54 
54 
54 

9-36336 
9-36394 
9.3645I 
9.36509 

9.36  566 

5? 

r 

0.63  663 
0.63606 
0.63  548 
0.63491 
0-63433 

9.98  872 
9.98869 
9.98866 
9.98  863 
9.98  866 

00 

59 
58 

i 

6 

5? 

5-? 

55 

5- 

r 
7 

5 

5 

6 

6 

56 
11 

:       1 

8 
9 

9-3548r 
9-35  536 
935590 
935644 
9-35698 

54 

54 

tt 

54 

9.36623 
9.36681 
9.36738 

9.36852 

57 
5? 

57 

0.63  376 
0.63319 
0.63  262 
0.63  204 
0.63  14? 

9.98858 
9.98855 
9.98852 

9-98849 
9.98846 

55 
54 
53 

8 
9 

10 
20 

6.7 

7-6 
8.6 
9.6 
19.1 

b. 

S. 

9- 
19. 

6 
6 

5 

5 

0 

6 

8 

9 

18 
28 

6 

•§ 
•  5 

-4 

6-5 
7-4 
8.4 
9-3 
1  8.0 
28  o 

10 

ii 

12 

13 
14 

9-35752 
9-35  ?06 
9.35866 

9-359H 
9.35968 

54 
54 
54 
53 
54 

9.36909 

9-36966 
9.37023 
9.37080 

9-37  136 

57 
57 
56 

i 

0.63  096 
0.63033 
0.62  977 
0.62  920 
0.62  863 

9.98843 
9.98  840 
9.98837 
9.98834 
9.98831 

50 

49 
48 
47 
46 

3° 
40 
50 

38.3 
47-9 

5$ 

38. 
47- 

5 

o 

5 

5 

57 
47 

5 

-6 
.1 

i 

37-3 
46-6 

54 

II 

17 

18 
19 

9.36021 
9.36075 
9.36123 
9.36  182 
9-3623? 

1 

53 
53 

c5 

9.37  193 
9.37250 

9.37306 
907363 
9.37419 

57 
56 
56 
56 

56 
cfi 

0.62  8og 
0.62  750 
0.62  693 
0.62637 
0.62  586 

9.98  828 
9.98825 
9.98822 
9.98  819 
9.98816 

45 

44 
43 
42 

6 

8 
9 

10 

S4 

7-4 

1 

8 
9 

•5 
4 

i 

1 

£ 

4 

!§ 

.2 
.1 

5-4 
6.3 
7-2 
8.1 

9.0 

20 

21 

22 
23 
24 

9.36  289 
9-36342 
9-3639! 
9-36448 
9.36501 

53 

53 
53 
53 
53 

r  -3 

9-374/5 
9-37  532 
9-37  588 
9-37644 
9.37706 

5° 
56 
56 

56 
56 

0.62  524 
0.62468 
0.62412 
0.62  356 
0.62  299 

9-98813 
9.98  816 
9.98807 
9.98  804 
9.98  80  1 

40 

39 
38 
37 
36 

20 
30 

40 

50 

18.5 
27.? 
37-0 
46.2 

18 
27 
36 
45 

-3 
•5 
.g 
J 

1  8 

36 

45 

.1 

•3 

•4 

27.0 
36.0 
45.0 

25 
26 
27 
28 
29 

9-36554 

9.  36  666 

936713 
9.36766 

53 

53 
53 
52 
53 

9-37  756 
9-378I2 
9.37868 
9-37924 
9-37  979 

56 
56 

55 

0.62  243 
0.62  1  88 
0.62  132 
0.62  076 
0.62  026 

9-98  798 
9.98795 
9.98792 
9.98789 
9:98  786 

35 
34 
33 
32 

6 
8 

5l 

*•! 

5. 

I 

7 

3 
3 

.2 

8 

- 

5 

( 
7 

2 
.2 
.1 

.O 

52 

5.2 
6.6 

6.9 

30 

32 
33 
34 

9.36813 
9.36871 
9.36923 
9.36976 
9-37023 

1 

52 
52 

9-3803^ 
9.38091 
9-38  146 

9.38  202 
9.3825? 

55 
55 
55 
55 

5? 

0.6  1  963 
0.61  909 
0.61  853 
0.61  798 
0.6  1  742 

9-98783 
9.98780 

9-98777 
9.98774 
9.98771 

30 

29 
28 

27 
26 

9 

10 

20 

30 
40 

o.O 

8.9 
17-8 
26.? 

35.6 

44  6 

8 

17 
26 

35 

44 

1 

8 
5 

f 

8 

17 
26 

35 

•9 

.0 

- 

7.8 
8-6 
17.3 
26.0 

34-6 
4-1.5 

11 

37 
38 
39 

9-37o8i 
9-37  133 
9-37  185 
9-37  23? 
9-37289 

52 

P 

52 

9-383I3 
9.38368 

9-38478 

9-38  533 

5 

55 
55 
55 

55 
ef 

0.61  687 
0.6  1  632 
0.61  576 
0.61  521 
0.6  1  466 

9.98768 
9-9876* 
9.98  762 

9-98759 
9.98755 

25 
24 

23 

22 
21 

l 

6 
7     < 

$ 

5-f 

' 
• 

5i 

5-  r 

v  '•  ' 

56 
5-6 
5-9 

40 

42 
43 
44 

9-37341 
937393 
9-37445 
9-37497 
9-37  548 

52 
52 
5? 

9.38  589 
9.38644 
9-38693 

9-38753 
9.38803 

55 

55 
54 
55 
55 

0.61  411 
0.61  356 
0.6  1  301 
0.6  1  246 
0.61  191 

9.98752 
9.98749 
9-98  746 
9-98  743 
9.98  746 

20 

19 
18 

17 
16 

8     ( 
9     ' 

10       < 

20   i; 
30  2 

5-8 
7-7 
3.6 

7.1 

>•? 

< 
i 

2 

'.  C* 

-•(': 

5-5 

7.0 

( 
i 

2 

5-? 
7.6 

3-4 

5.8 

5-2 

45 
46 
47 
48 

49 

9.37606 
9.37652 
9-37703 
9-37755 
9-37806 

1 

5i 

9.38  863 
9.38918 
938  972 
9.39027 
9.3908! 

54 

i 

54 

0.61  137 
0.61  082 

0.6  1  02? 

0.60  973 
0.60  913 

9-98737 
9-98734 
9.98731 
9.98728 
9.98725 

15 
14 
13 

12 
II 

40   3^ 
50  4 

1-3 
2.9 

3 

3- 
4 

4-  .  O 

-0 

3 

3 
4 

3-6 

2.  1 

50 

52 
53 

54 

9.3785? 
9.37909 
9-3796o 
9.38011 
9.  38  062 

51 

$1 

51 

51 

51 

9-39  136 
9-39  190 

9.39299 
9-39353 

54 
54 

54 
54 
54 

0.60  864 
0.60809 
0.60755 
0.60  701 
0.60  647 

9.98721 
9.98718 
9-98/I5 
9.98712 
9.98  709 

10 

9 
8 

6 

6 

1 

9 

10 

0-3 
0.4 
0.4 
0.5 
0.6 

0 
0 
0 
0 
0 

•3 
•3 
•4 

4 

•5 

0. 
G. 

0. 

3 
3 

1 
J 

P 
P 

59 

9-38113 
9.38  164 
9.38215 
9.38  266 
9-383I7 

5r 

5i 
51 

rA 

9-3940? 
9.3946T 
9.395I5 
9.39569 
9.39623 

54 
54 
54 
54 
54 

0.60  592 
0.60  538 
0.60484 
0.60436 
o.  60376 

9.98706 
9.98  703 
9.98  700 
9.98696 
9.98693 

5 
4 
3 

2 
I 

30 
40 
50 

2^3 

2.9 

I 

2 
2 

•  5 

.0 

•  5 

O. 

I. 
I. 

2. 

j 

1 
i 

60 

9.3836? 

5° 

9-39677 

53 

0.60  323 

9.  98  696 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

' 

p. 

1 

361 


TABLE  VII. —LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

14° 


Log.  Sin. 

d. 

Log.  Tan. 

c.  d.     Log.  Cot. 

Log.  Cos. 

d. 

p.  p. 

o 

9.38418 
9.38463 

9.38519 
9.38569 

56 
56 
50 
56 
50 
50 
50 
50 
50 
50 
50 
50 
50 
50 

50 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
48 
48 
49 
48 
48 
48 
48 
48 

48 
48 

48 
48 
48 
48 
48 
48 
4? 
48 
4? 
4? 
4? 
4? 
4? 
4? 
4? 
47 
4? 
47 

9.39677 
9.39731 
9.39784 

9.39892 

54 
53 
54 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
53 
52 
53 
52 

53 

52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
52 
5? 
52 
52 

11 

5* 

i? 

51 
5? 
51 

5* 

51 

5i 
56 
51 
56 
56 
56 

0.60323 
O.6o  269 
0.6o2l5 
0.60  161 
0.60  1  08 

9.98  696 
9.9868? 
9.98684 
9.98681 
9.98678 

3 

3 
3 
3 
3 
3 
3 

3 

3 

3 

3 

3 

3 

3 

3 

3 
3 
3 

3 
3 

3 

3 

3 
3 
3 

3 
3 

3 

3 
3 

3 

3 

3 

3 
3 

3 

3 
3 

3 

3 

3 

3 

3 
3 

3 

3 

60 

P 

i 

( 

« 

1 
K 

2< 

3< 
4< 
5< 

6 

8 
9 

10 
20 
30 
40 

50 

6 

8 
9 

10 
20 
30 
40 
50 

6 

8 
9 

10 

20 
30 
40 
50 

5' 

5     5- 

7      6. 

*     7- 
?     8. 

3       9- 
D     18. 
D    27. 
3    36. 

>  45- 

55 

5-2 
6.1 
7.0 

7-9 
8.? 

17.5 
26.2 

35-o 
43-? 

56 

5.6 

7.6 

8.4 
16.3 

25.2 

33-6 
42.1 

48 

4-8 
5-6 
64 
7.3 
8.1 
16.1 
24.2 

32-3 
40.4 

6 

8 
9 

10 
20 
30 
.  40 
50 

I        f 

4     5 
3    *• 

2       7 

O      i 
O    I/ 
0    2t 

o  3  = 
o  4^ 

52 

5.2 
6.6 
6.9 

174 
26.0 

34-6 
43-3 

50 

5-o 

5-8 
6.6 

16.5 

25.0 
33-3 
41-6 

48 

4-8 

I'6 
6.4 

7.2 
8.0 
16.0 
24.0 
32.0 
40.0 

3 

0-3 
0.4 

0.4 

o.5 
0.6 
i.i 
i.? 
2-3 
2.9 

3 

•  3 

!i 

.0 

'.9 

-8    i 

>.?     2 

•6   3 
^.6  4 

6!o 
6.8 
7-7 
8.6 
17.1 
25-? 
34.3 
42.9 

49 

4-9 

Ii 
tt 

24.? 
33.0 
41.2 

4? 

4-? 

S4 

7.1 

7-9 
15-8 

3*2 

39-6 

3 

0.3 
0-3 
0.4 
04 
0.5 

I.O 

2.0 

2-5 

53 

7.6 
7-9 
8.8 
7-6 
6-5 
5.3 
4-1 

5-9 
6.8 

7-6 
8.5 
17.0- 

25-5 
34-0 
42.5  i 

49 

4-9 

11 

7-3 
8.1 

16.3  ! 

24.5  i 

32.6 

40.3 

47 

4.7 

n 
7.6! 

7-8! 

15-6  ! 
23.5 
3L3 

39-  ? 

6 

9 
ii 

12 

13 
14 

9.38  620 
9.38676 
9.38726 
9.38771 
9.38821 

9-39945 
9-39999 
9.40052 

9.40  1  06 
9.40159 

0.60054 
0.60001 
0.5994? 

o.  59  894 
0.59841 

9.98674 
9.98671 
9.98663 
9.98665 
9.98662 

55 
54 
53 
52 

9.38871 
9.38921 
9.38971 
9.39021 
9.39071 

9.40212' 
9.40265 
9-403I8 
9-40372 
9.40425 

0.5978? 

0.59734 
0.59  68T 
0.59628 
0-59575 

9.98653 
9.98655 
9.98652 
9.98  649 
9.98  646 

50 

49 
48 

47 
46 

15 

16 

18 
19 

9.39126 
9.39176 
9.39220 
9.3926§ 
9-393I9 

9.40478 
9.40531 
9.40  583 
9.40635 
9.40689 

0.59522 
0.59469 
0.59415 

0.59363 
0.59311 

9.98642 
9.98639 
9.98635 
9-98633 
9.98  630 

45 

44 
43 
42 

20 

21 

22 
23 
24 

9.39418 

9-395l£ 
9.39566 

9.40742 

9.40794 
9.40847 
9.40  899 
9.40952 

0.59258 
0.59205 

0.59153 
0.59  106 
o.  59  048 

9.98  625 

9.98  620 
9.98617 
9.98613 

40 

39 
38 

1 

25 

26 

27 
28 

29 

9.39615 
9.39664 

9-397I3 
9-39762 
9.398II 

9.41  004 
9.41  057 
9.41  109 
9.41  161 
9.41213 

0.58995 

0.58943 
0.58  891 
0.58833 
0.58785 

9.98  616 
9.98  607 
9.98604 
9.98  606 
9.98  59? 

35 
34 
33 
32 

30 

32 
33 
34 

9.  39  860 
9-39909 
9-3995? 
9.40006 
9.40055 

9.41  266 
9.41  318 
9.41  370 
9.41  422 
9.41  474 

0.58734 
0.58682 
0.58  630 
9.58578 
0.58  526 

9.98  594 
9.98  591 
9.98  58? 
9.98  584 
9.98  581 

30 

29 
28 

27 
26 

35 
36 

39 

9.40  103 
9.40  152 

9.40  200 
9.40249 
9.40297 

9.41  525 

9-4i  57? 
9.41  629 
9.41  681 
9.41  732 

0.58474 
0.58422 
0.58376 
0.58319 
o.  58  26? 

9.98  578 
9.98  574 
9.98571 
9.98  568 
9.98  564 

25 
24 

23 

22 
21 

40 

42 
43 
44 

9-40  34? 
9.40  394 
9.40442 
9.40490 
9.40  538 

9.41  784 
9.41  836 
9.41  88? 

9-41  938 
9.41  990 

0.58216 
0.58  164 

0.58  112 
0.58061 

0.58  cio 

9.98  561 

9.98  558 
9.98  554 
9.98  551 
9.98  548 

20 

19 
18 

3 

4I 
46 

47 
48 

49 
50 

52 
53 

54 

9.40  586 
9.40634 
9.40  682 
9.40  730 

9.4077? 

9.42  041 
9.42092 

9-42  H4 
9.42  195 
9.42  245 

0.57958 
0.5790? 
0.57856 
0.57805 
0-57753 

9.98544 
9.98  541 
9-98  538 
9-98  534 
9.98  531 

15 
14 
13 

12 
II 

9.40825 
9.40  873 
9.40926 
9.40  968 
9.41015 

9.42  29? 
9-42  348 
9.42  399 
9.42  456 
9.42  501 

0.57702 
0.57651 
o.  57  606 
0.57549 
0.57499 

9.98  528 
9.98  524 
9.98  521 
9.98  518 
9.98  514 

10 

6 

on  On  On  On  on 

VO  OOVI  ONOn 

9.41  063 
9.41  1  16 
9.41  158 
9.41  205 
9.41  252 

9.42  552 
9.42  602 
9.42  653 
9.42  704 
9.42  754 

o.  57  448 
0-5739? 
0.57346 
0.57  296 
o  57  245 

9.9851! 
9.98  508 
9.98  504 
9.98  501 
9.98498 

5 
4 
3 

2 

I 

00 

9.41  299 

9.42  805 

0-57  195 

9.98494 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

' 

P.  P. 

7JV 


TABLE  VII.  —  LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

15° 


1 

Log.  Siu. 

d. 

Log.  Tan.  !  c.  d. 

Lo?.  Cot. 

Log.  Cos.        d. 

p.  p.               ! 

0 

I 

2 

3 

4 

9.41  299 

9-41  346 
9-41  394 
9.41  441 
9.41  488 

47 
tf 

47 
47 
46 
47 
47 
46 
46 
47 
46 
48 
48 
48 
46 
48 
48 
46 
46 

4£ 
46 

46 
45 
46 
46 
46 
4! 

4I 
46 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
44 
45 
44 

*| 

44 
44 

45 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 

9.42  805 
9.42  856 
9.42906 
9.42  956 
9.43007 

51 
50 
56 
50 
50 
50 
50 
50 
50 
50 

5° 
50 
5o 
5o 
49 
50 
49 
49 
50 

49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
49 
48 
49 
49 
48 
49 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 
48 

48 
48 
48 
48 
48 

4? 
48 
48 

4f 

48 

49 

0.57  195 
0.57  144 
9.57094 

0.57043 
0.56993 

9.98494 
9.98491 
9-9848? 
9.98484 
9.98481 

3 

3 
3 

3 
3 
3 

1 

3 
3 
3 
3 

3 
3 

3 

3 

i 

3 
3 
3 
3 

3 
3 
3 
3 

! 

3 

3 
4 

3 

3 
4 
3 
3 

4 
3 

4 

00 

ii 
i 

6 

8 
9 

JO 

20 
30 
40 
50 

6 

8 
9 

10 

20 

30 
40 

50 

6 

I 

9 

10 
20 
30 
40 
50 

6 

8 
9 

10 
20 
30 
40 
50 

49 

4.9 

1:1 

7-4 

8.2 

16.5 
24-? 
33-c 
41.5 

4? 
tf 

*! 

7-i 
7-S 

is-l 

23-7 
3i-e 
39-^ 

45 
4-1 

8 

6.* 

7-< 

15.1 
22.5 
30.; 
37-S 

6 

7 
8 

9 

10 
20 
30 
40 
50 

5° 

5-6 
5-9 
6-? 
7.6 
8.4 
i6.§ 
25.2 

33-8 
42.1 

49 

4-9 

t\ 

8.1 
16.3 
24.5 
32.8 
40-8 

47 

4-7 
5-5 
6.2 
7.6 

>   7-8 

15-8 
23-5 

:£? 

45 

'   4-5 

;  5-2 

)   6.0 
5   6.? 

>    7-5 
15.0 

'22.  5 

J30.o 
>37-5 

4 

0.4  c 
0.4  c 
o.5   c 
0.6   c 
0.8  c 

2.0     I 

2.8   i 
3-3    2 

50 

5-o 

5-8 
6-6 

1:1 

i6.£ 

25.C 

33-2 

4i.e 

48 

4-8 
5-8 
6.4 

S3 

16.1 
24.2 

32-3 
40.4 

46 

4-8 
5-4 

6.2 

7-o 
7-? 
15-5 
23-2 
31.0 
38.? 

44 

4.4 

P 

6-7 
7-4 
14-8 

22.2 
29-8 
37-1 

3 

'•3   c 
>.4   c 
».4   c 

'•5   c 
>.6   c 

.T    i 

•?    i 
•3   2 
•9   ^ 

1 

> 

48 

4-8 
5-6 
6-4 
7-2 
8.0 
16.0 
24.0 
32.0 
40.0 

46 

4-6 

5-3 
6.T 

6-9 

7-8 
15-3 
23.0 

30-8 
38.3 

44 

4.4 

11 

6.6 
7-3 
14-8 

22.0 
29-3 
36.8 

3 

'-3 
>.  3 

>-4 
>-4 
>-5 

.0 

•5 
.0 

-5 

i 

8 
9 

9-4i  534 
9.41  581 
9.41  623 
9.41  675 
9.41  721 

9-43  °5? 
9-43  I0? 
9.43I57 
9.43  208 
9.43258 

o.  56  942 

0.56892 

o.  56  842 
o.  56  792 
o.  56  742 

9.9847? 
9.98474 
9.98476 
9.98467 
9.98464 

55 
54 
53 
52 
5i 

10 

ii 

12 

13 
i      14 

9  41  768 
9.41  815 
9.41  861 
9.41  908 
9.41  954 

9-43308 

9-43  358 
9.43408 

9-43  45s 
9-43  5°8 

o.  56  692 
o.  56  642 

0.56592 
0.56542 

o.  56  492 

9.98466 
9.98457 
9-98453 
9.98  45° 
9.98  448 

50 

.49 

48 
47 
46 

15 

16 

17 

18 

19 

9.42006 
9.42  047 
9.42093 
9-42  139 
9.42  1  8^ 

9-43557 
9.43607 

9.43657 
9-43  706 
9-43758 

0.56  442 

0.56392 

0.56343 
0.56293 
0.56243 

9-98443 
9-98439 
9-98436 
9-98433 
9.98  429 

45 
44 
43 
42 
4i 

20 

21 

22 

23 

1      24 

9.42  232 
9.42  278 
9.42  324 
9.42  369 
9.42415 

9-43  806 
943855 
9-43905 
9-43  954 
9.44003 

0.56  194 
0.56144 
0.56095 
0.56045 
0.55998 

9.98426 
9.98422 
9.98419 
9.98415 
9.98  412 

40 

39 
38 

I 

a! 

27 
28 
29 

9.4246! 
9.42  50? 
9-42  553 
9-42  598 
9.42644 

9-44053 
9.44  102 
9.44151 
9.44206 
9-44249 

0-55947 
0.55898 
0.55843 
0.55799 
0.55756 

9-98403 
9.98405 
9.98401 
9.98  398 
9-98  394 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 
34 

9.42  690 
9-4273S 
9.42781 
9.42  825 
9.42  871 

9.44299 
9-44  348 
9-44  397 
9.44446 

9-44494 

0.55  701 
0.55652 
0.55603 

0-55554 
0.55  505 

9.98  391 
9.98  387 
9.98  384 
9.98  386 
9.98  377 

30 

29 
28 
27 
26 

P 
% 

39 

9.42917 
9.42  962 

9-430°? 
9.43052 
9.43098 

9-44  543 
9-44  592 
9.44641 
9.44690 
9-44738 

0-55456 
o.554o? 

0-55359 
0.55310 
0.55  261 

9-98373 
9.98  370 

9-98  368 
9-98  363 
9-98  359 

25 
24 

23 

22 
21 

40 

4i 
42 
43 
44 

9-43  H3 
9-43  188 
9-43233 
9.43278 

9-43  322 

9.44787 

9.44835 
9.44884 

9-44932 
9.44981 

0.55213 

0-55  l64 
0.55  116 
0.55067 
0.55019 

9.98  356 
9.98352 
9-98348 
9-98  345 
9.98  341 

20 

19 
18 

17 
16 

45 
46 

47 
48 

49 

9-43  36? 
9.43412 

9-43457 
9.43  50! 

9-43  546 

9-45  029 
9-45  °77 
9.45  126 
9.45  174 
9.45  222 

o.  54  976 
0.54922 

0.54874 
0.54825 

0.54777 

9.98  338 
9-98  334 
9-98  33i 
9.98  32? 
9.98  324 

15 
14 
13 

12 
II 

50 

5i 
52 
53 
54 

9-43  59i 
9-43635 
9.43680 

9-43724 
9-43768 

9-45  276 
9-45318 
9-45  367 
9.45415 

9-45463 

0.54729 
0.54681 

0.54633 
0.54585 

o-  54  537 

9.98320 
9-98318 
9-983I3 
9.98309 

9-98  306 

10 

9 

8 

7 
6 

g 
9 

59 

9-438I3 
9-43  857 
9-43  901 
9-43945 
9-43  989 

9.45516 

9-45  558 
9.45  608 

9.45654 
9-45  702 

o.  54  489 
0.54441 

0-54393 
0.54346 
o.  54  298 

9.98  302 
9.98293 
9.98  295 
9.98  291 
9.98288 

5 
4 
3 

2 

I 

60 

9.44034 

9-45  749 

0.54256 

9.98  284 

0 

Log.  Cos.        (1. 

Log.  Cot. 

c.  d.  i    Log.  Tan 

Log.  Siii.         d. 

/ 

P.  P. 

74' 


363 


TABLE  VII. —  LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS 

16° 


t 

Log.  SI  a. 

d. 

Log.  Tan.  c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p 

.  P. 

0 

I 

2 

3 
4 

9.44034 
9.44078 
9.44122 

9.44  1  66 
9.44209 

44 
44 
44 
43 
44 
44 
43 
43 
44 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

43 
42 

43 
43 

42 

42 

43 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
42 
4t 
42 
42 

4i 
42 
4i 
41 

42 

4f 
4* 

4? 
41 
4i 
4t 

9-45  749 
9-45  797 
9.45  845 

9-45  892 
9.45  940 

48 
4? 
4? 
4? 

4? 
4? 
4? 
47 
47 
47 
4? 
47 
47 
47 
47 
47 
47 
47 
46 
47 
47 
46 
46 
47 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

46 
46 
46 

46 

46 

46 

46 
45 

46 
46 

45 

46 

45 

46 

45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 
45 

0.54250 

o.  54  202 

0.54155 
0.5410; 

o.  54  060 

9.98  284 
9.98  286 
9.98  277 
9.98273 
9.98  269 

3 

3 
4 
3 

4 
3 

4 
3 

4 
3 
3 
4 

3 

4 
3 
4 

1 

4 
3 
4 
3 

4 
3 

4 
3 

4 
3 
4 
3 
4 
3 
4 
3 
4 
3 
4 
4 
3 
4 
3 
4 
4 

4 
4 

4 
4 
3 
4 
4 

4 
4 

GO 

59 
58 

I 

6 

7 
$ 

9 

10 
20 
30 
40 

50 

6 

7 
8 

9 
10 

20 
30 

4°. 

50. 

6 

8 
9 

10 

20 
30 
40 
50 

6 

8 
9 

10 

20  J 
30: 

402 
50|; 

4* 

4- 

1: 

S. 
1  6. 

24. 
32. 
40 

46 

4-6 

5-4 

6.2 

7-o 
7-? 
15-5 
23.2 
31.0 
38.? 

4' 

4 
5- 

!: 

7. 
14. 

22. 

29- 

36. 
42 

4-2 
4-9 
5-6 
6-4 
7-i 
4.1 

51.2 

-8.3 
15-4 

6 

I 

9 

10 

20 
30 
40 
50 

5 

8 
6 
4 

2 
O 
0 
0 

o 

0 

4 
i 
i 
I 

( 

1  = 

2^ 

3^ 
3^ 

I 

4 
i 

8 
6 

i 

0 

4 

4 
4 

( 

7 
M 

21 

28 

35 

c 
c 
c 
c 
c 
I 

2 
2 

3 

4 

4 

I 

7 
7 

15 

22 

3i 
35 

6 

[.6 

J-3 
).! 

>-9 
>4 

»-3 

)-0 

'•6 
-3 

4 

4 

c 

< 

IH 
21 
2C 

3< 

2 

.2 

•9 
.6 

•3 

.0 

.0 
.0 

.0 
.0 

t 

•4 
•4 

:I 

* 

.0 

1 

•  7 
•3 

5 

•7 
•6 
.6 

4 

4 

6 
6 
7- 

15 

22 
30 

37 
3 

1-3 

•  i 

.8 
>o 

.2 

L-5 
).o 

.2 
4' 

4- 
4- 

6! 

6. 

T3- 
20. 

27. 

34- 

3 

o. 

0. 

o. 

0. 
0. 

1. 
I. 

2. 

2. 

I 

2 

3 
3 

5 
5 
3 
6 
8 
6 
I 

7 
3 
9 

i 
* 

2 

3 

T 

8 

2 

9 

6 
6 

3 

4 
4 

1 

T 

7 

9 

47 

4-7 

1:1 

7.6 

7-8 
5.6 
3-5 

<-3 

9.i 

45 

4-5 

6!o 

6-? 

7-5 
15.0 

22.5 
30.0 

37-5 

43 

4-3 

8 

64 
7-t 
4-3 
i-5 
8.6 

5-8 

4i 
4.1 
4-8 

54 
6.1 

6.8  ! 
13-6 
20.5 

27.3! 

34-  1  , 

6 

8 
9 

9.44253 
9.44  29? 

9-44341 
9.44  384 

9-44428 

9-45  98? 
9.46035 
9.46  082 
9.46  129 
9.46  177 

0.54012 
0.53965 

o.539i? 
0.53876 
0.53823 

9.98  266 
9.98262 

9.98  258 
9.98255 
9.98251 

55 
54 

53 
52 
5i 

10 

ii 

12 
13 
H 

9.44472 
9.44515 

9-44  559 
9.44602 
9.44646 

9.46  224 
9.46  271 

9-46318 
9.46  366 
9.46413 

•0.53776 

0-53728 
0.53  68T 

0.53634 
0.53587 

9.98247 
9.98  244 
9.98  246 
9.98  235 
9-98233 

50 

49 
48 
47 
46 

II 

17 

18 
'9 

9.44689 
9.44732 
9.44776 
9.44819 
9.44  862 

9.46  460 
9.46  507 
9.46  554 
9.46  60  1 
9.46  64? 

o.5354o 

0-53493 
0.53446 

0-53399 
0-53  352 

9.98  229 
9.98225 

9.98  222 
9.982I8 
9.98  214 

45 
44 
43 

42 

4i 

20 

21 

22 

23 
24 

9.44905 
9-44  948 
9.44991 

9-45  034 
9.45  07? 

9.46  694 
9.46  741 
9.46  788 
9.46  834 
9.4688! 

o.533o5 
0-53258 
0.53212 
0.53165 
0-53  H8 

9.98  211 
9.98  207 
9.98  203 
9.98  200 
9.98  196 

40 

39 
38 
37 
36 

25 
26 

27 
28 
29 

9.45  120 

9.45  163 
9.45  206 

9-45  249 
9.45  291 

9.46  928 

9-46  974 
9.47021 
9-47  067 
9-47  H4 

0.53072 
0.53025 
0.52979 
0.52932 
0.52886 

9.98  192 

9.98  i8§ 

9.98  185 

9.98  181 
9.98  17^ 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 
34 

9-45  334 
9-45  377 
9.45419 
9.45  462 
9.45  504 

9.47  1  66 
9-47  207 
9-47  253 
9.47  299 
9-47  345 

0.52839 
0.52793 

0.52747 
0.52  706 
0.52654 

9-98173 
9.98  170 
9.98  1  66 
9.98  162 
9.98158 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9-45  547 
9.45  589 
9-45631 
9-45  674 
9.45716 

9-47  392 
9-47  438 
9-47  484 
9-47  530 
9-47  576 

0.52  608 
0.52  562 
0.52  516 

0.52469 
0.52423 

9.98  155 
9.98151 
9.98  14? 
9.98  143 
9.98  140 

25 
24 

23 

22 
21 

40 

4i 

42 
43 
44 

9-45  758 
9.45  800 

9-45  842 
9-45  885 
9-45  927 

9.47  622 
9-47  66§ 
9-47  7H 
9-47  76o 
9.47  806 

0.52377 
0.52331 
0.52  286 
0.52  240 
0.52  194 

9.98  136 
9-98  132 
9.98  I2§ 
9.98  124 

9.98  121 

20 

19 
1  8 

17 
16 

45 
46 
47 
48 

49 

9.45  969 
9.46011 
9.46052 
9.46094 
9-46  136 

9-47  85! 
9-47  897 
9-47  943 
9.47989 
9.48034: 

0.52  H8 

0.52  I  O2 
0.52057 

0.52  on 

0.51  965 

0.51  920 

0.51  874 

0.51  829 

0.51  783 
0.51  738 

9.98  117 
9.98113 
9.98  IO§ 

9.98  \o\ 

9.98  102 

15 
H 
13 

12 
II 

50 

5i 
52 
53 
54 

9.46  178 

9.46  220 
9.46  26f 
9.46  303 

9-46  345 

9.48  080 
9.48  125 
9.48  171 
9.48215 
9.48  262 

9.98  098 
9.98094 
9.  98  096 
9.98085 
9.98082 

10 

9 
8 

{ 

55 
56 

57 
58 
59 

9-46  386 
9.46428 
9.46469 
9.46511 
9-46552 

9-48  30?  . 

9-48353 
9.48  398 
9.48  443 
9.48  488 

o.  5  1  692 
0.51  647 
o.  5  1  602 

0.51  556 
0.51  511 

9.98079 
9.98075 
9.98071 
9.9806? 
9.98063 

5 
4 
3 

2 

I 

60 

9-46  593 

9-48  534 

o.  5  1  466 

9.98059 

0 



Log.  Cos. 

d. 

Log.  Cot. 

c.  d.   Log.  Tan. 

Log.  Sin. 

d. 

/ 

p 

.  p. 



364 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

17° 


/ 

LOST.  Siii.        <]. 

I.<>ir.  Tan. 

c.  d.     Log.  Cot. 

Lop.  Cos. 

d. 

p. 

p. 

0 

I 

2 

3 
4 

9-46  593 
9.46  635 
9.46676 
9.46717 
9-46758 

41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
41 
46 

41 

46 
46 

41 
46 
46 
40 
46 
46 
40 
46 
46 
40 
46 
40 
43 
40 

4? 
40 
40 
40 
40 
40 

39 
40 

39 
40 

39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
39 
29 
39 
38 

9-48  534 
9.48  579 
9.48624 
9.48669 
9.48714 

45 
45 
45 
45 
45 
45 
44 
45 
45 
45 
44 
45 
44 
44 
45 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
44 
41 
41- 
43 
44 
44 
44 
43 
44 
43 
44 
43 
43 
44 
43 
A3 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 
43 

o.  5  1  466 
o.  5  1  42  1 
0.51  376 
0.51  336 
0.51  285 

9.98059 
9.98056 
9.98052 
9.98048 
9.98044 

3 
4 
4 
4 
3 
4 
4 
4 
4 

3 
4 

4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
4 
3 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 

60 

It 

I 

6 

7 
8 

9 

10 
20 

30 
40 

50 

6 

8 
9 

1C 

2C 

3^ 

40 
50 

I 

2 

3 

4 
5 

45 
4.1 

d 

6.i 

?.<• 
I5.i 

22.^ 

3°-' 
37-< 

6 

i 

9 

10 

20 
30 
40 
50 

:' 

4- 
5- 
6. 
6. 
13- 

20. 

34- 

6 

7 
8 

9 
o 

0    I 
0     I 

0    2 

t>!3 

6 

8 
9 

10 
20 
30 
40 
50 

45 

;  4. 

It: 

5   6. 

>   7- 

> 

r  22. 

53°. 
?37 

4 
4 
5 

I 

7 

H 

21 

29 
36 

4 

i    4 
8    4 

5    5 

2     6 

9   6 
813 

?;2C 

627 

6|34 

39 

3i 
4.6 

5-2 

5-9 
6.6 

$ 

6-3 
2.9  1 

4 

0.4 
0.5 
0.6 

0.7 

0.? 

'•s 

2.2 

i? 

5 

2 
0 

7 

s 

o 
S 

0 

s 

3 

5 
i 

8 
5 

2 

5 

o 

2 

I 
.  I 

.8 

4 
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•S 
•3 
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3 

1 

i 
i 

': 

i< 

2( 

3- 
o 

0 

c 
0 

o 

I 

2 
2 

3 

44 

4-4 

u 

6.7 
7-4 
14.* 

22.: 

29-e 
37.1 

4 

4 

r 

6 

7 
H 

21 

28 

35 

4c 

4- 
4- 

I: 

6. 
13- 

20. 
27. 

33- 

9 

5-9 

L-5 

5.2 

58 
>-5 
5-0* 
>-5 

).O 
5-5 

4 

-4 
•  4 

!e 

•6 
•3 

.0 

•6 
•3 

,  44 
-   4-4  ' 
5-t 
>    5-8 
6.6 

L  7.3 
;i4-6 

\  22.  0 

i!29-3 
136-S 

3 

3 
o 

4 
i 

-3 

-6 
-8 

40 

6   4.0 
7    4-6 
4   5-3 

I     6.0 

J  -6-6  ! 
513.3, 

2  2O.O 
0  26.6 
?  33-3  i 

38 

3.8 
4-5 

1:1 

6-4 

I2.§ 

19.2 

25-6 
32.1 

3 

0.3 
0.4 
0.4 

o-5 
0.6 
i.l 

2'.  3 
2.9 

1 

8 
9 

9.46  799 
9.46  846 
9.46  881 
9.46922 
9-46963 

9.48759 
9.48  804 
9.48  849 
9.48  894 
9.48939 

0.51  240 
0.51  195 
0.51  151 
0.51  106 
0.51  06  1 

9.98046 
9.98036 
9.98032 

9.98[o2§ 
9.98024 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 
14 

9.47004 
9.47  04^ 
9-47  085 
9-47  127 
9.47  1  68 

9.48  984 
9.49023 
9-49073 
9.49  118 
9.49  162 

0.51  016 
0.50971 
o.  50  925 
0.50882 
0.50837 

9.98021 
9.98017 
9.98013 
9.98009 
9.98005 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

9.47  203 

9-47  249 
9.47  290 

9-47  330 
947371 

9-49  207 
9-49252 
9-4929S 
9-49  34i 
9-49385 

0.50792 
o.  50  748 
0.50703 
0.50659 
0.50614 

9.98001 
9-9799? 
9-97993 
9.97989 
9.97985 

45 
44 
43 
42 
4i 
40~ 
39 
38 

i 

20 

21 
22 
23 
24 

9.47411 
9.47452 
9-47  492 
9-47  532 
9-47  573 

9-49430 
9-49474 
9-49518 
9-49  563 
9-496o? 

0.50570 
0.50525 
0.50481 
o.  50  437 
o.  50  392 

9.97  981 
9-97  97? 
9-97973 
9.97969 
9-97  966 

26 
27 
28 
29 

9.47613 
9-47  653 
9.47694 

9-47  734 
947774 

9.49651 

9-49695 
9-49740 
9.49784 
9.49  828 

o.  50  348 
0.50304 
o.  50  260 
0.50216 
0.50  172 

9-97  962 
9.97958 

9-97954 
9.97950 
9.97946 

35 
34 
33 
32 
3i 

30 

3i 
32 
33 
34 

9-47  814 
9.47  854 
9.47  894 
9-47  934 
9-47  974 

9.49  872 
9.49  9i§ 
9.49960 
9.50004 
9.  50  048 
9-50092 
9.50136 
9.50179 
9.50223 
9.50267 

0.50  128 
o.  50  083 
o.  50  039 

0.49  996 
0.49952 

9-97  942 
9.97938 
9.97934 
9-97930 
9-97  926 

30 

29 
28 
27 
26 

35 

36 

% 

39 

9.48014 
9.48054 
9.48  093 
9-48  133 
9-48  173 

0.49  908 
0.49  864 
0.49  826 
0-49  776 

0.49  733 

9.97922 
9.97918 
9.97914 
9.97910 
9-97  906 

25 
24 
23 

22 
21 

40 

4i 
42 
43 
1    44 

45 
46 
47 
48 

49 

9.48213 
9.48  252 
9.48292 
9-48  33i 
9-4837I 

9.50311 
9.50354 
9-  5°  398 
9.  50  442 
9.5048^ 

0.49  689 
0.49  645 
0.49602 
0.49  558 
0.49514 

9.97902 
9.97898 
9.97894 
9.97890 
9.97  886 

20 

19 
18 

17 
16 

9.48416 
9.48450 
9.48489 
9.48  529 
9.48  568 

9.50529 
9-50572 
9.50616 
9.50659 
9.  50  702 

0.49471 
0.49427 
0.49  384 
0.49346 
0.49  297 

9.97  881 
9.9787? 

9.97873 
9.97869 
9.97865 

15 
H 
13 

12 

II 

50 

5i 
52 
53 
54 

9.48607 
9.48645 
9,48686 
9.48  725 
9.48  764 

9-  5°  746 
9.50789 
9.50832 
9.50876 
9.50919 

0.49254 
0.49216 
0.49  1  6? 
0.49124 
0.49081 

9.97861 
9.9785? 

9.97853 
9.97849 

9-97  845 

10 

1 

7 
6 

P 

* 

59 

9.48  803 
9.48  842 
9.48  881 
9.48  926 
9.48959 

9.50962 
9/51005 
9.51  048 
9.51091 
9-51  134 

0.49  038 
0.48994 
0.48951 
0.48903 
0.48  865 

9.97841 
9-97837 
9-97  833 
9.97829 
9.97824 

5 
4 
3 

2 

I 

«0 

9.48998 

9.51  17? 

0.48822 

9.97  826 

0 

Log.  Cos. 

d. 

LOST.  Cot.     c.  d.      Log.  Tan. 

Log.  Sin.    i      d. 

/ 

1' 

P. 

72' 


365 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

18° 


/ 

Loic.  (Sin. 

d. 

Lot?.  Tan. 

c.  d. 

Lot?.  Cot. 

Log.  Cos. 

d. 

p 

I'. 

0 

I 

3 

4 

9.48  998 

9-49037 
9.49  076 

9.49114 
9.49153 

39 
39 
38 
39 

9.51  17? 
9.51  220 
9.51  263 

9.51  306 
9-51  349 

43 
43 
43 
43 
/i3 

0.48822 

0.48  779 
0.48  736 
0.48  693 
0.48  656 

9.97  826 
9.97816 
9.97812 
9.97803 
9-97  804 

4 
4 
4 
4 

00 

59 
58 

| 

6 

4: 

4- 

5 
3 

4 
4 

2 

2 

42 

4.2 

6 

8 
9 

9-49  192 
9.49231 
9.49269 
9.49308 
9-49  346 

38 
39 
38 
38 
38 

•70 

9.51  392 
9.51435 
9.5I47? 
9.51  526 

9-5i  563 

42 

43 

42 

43 

42 

0.48  608 
0.48  565 
0.48  522 
0.48  479 
0.48  437 

9.97  800 
9.97796 
9.97792 
9-97  78? 
9-97  783 

4 

4 
4 

.4 
4 

55 
54 
53 
52 
5i 

8 
9 

10 

20 

5- 

6. 

7- 
14. 

0 

7 
4 
I 

3 

4 

6 
7 
14 

9 
6 

4 

T 
a 

4-9 

5.6 

6-3 
7.0 
14.0 

10 

ii 

12 
13 

H 

9-49385 
9-49423 
9.49462 

9-49  5°5 
9-49  539 

38 
38 
38 
38 
38 
1% 

9.51605 
9.51643 
9.51691 

9-51  733 
9.51  776 

42 

43 
42 

42 
42 

/19 

0.48  394 
0.48  351 
0.48  309 
0.48  265 
0.48  224 

9-97779 
9-  97  77  S 
9.97771 
9.97767 
9-97  763 

4 
4 
4 
4 

7 

50 

49 
48 

47 
46 

3° 

40 
50 

28. 
35- 

5 
6 
8 

/] 

28 
35 

f 

3 
4 

A 

28.0 
35-o 

j$ 

15 

16 

17 
18 

19 

9-49  577 
9.49615 

9.49653 
9.49692 

9-49730 

3° 
38 
38 
38 
3^ 

oQ 

9.51813 
9.51  861 
9.51903 
9.51946 
9.51  988 

42 

42 
42 

42 

42 

0.48  181 
0.48  139 
0.48  095 
0.48  054 

0.48  OI2 

9-97758 
9-97754 
9.97756 

9-97  746 
9.97742 

4 
4 
4 
4 
4 

45 
44 
43 
42 
4i 

6 

7 

8 

9 

10 

A 

A 

: 

e 

c 

k* 

•8 

-5 

.2 
.0 

l 
± 

1 
( 

M 
rf 

5-4 
S.i 

5-8 

20 

21 
22 

23 
24 

9-49  7b<> 
9.49  805 

9-49  844 
9.49  882 
9.49920 

J5 

38 
38 
38 
38 

•28 

9.52030 
9.52073 
9.52  115 

9.52157 
9  52  199 

2 

42 
42 

42 
42 

0.47  969 
0.47  927 
0.47  885 
0.47  842 
0.47  806 

9-97737 
9-97733 
9-97  729 
9.97725 
9.97721 

4 
4 
4 
4 

40 

3 
j 

1 

zo 

30 
10 

JO 

I; 

2C 

27 

34 

•8 
).? 

-6 
.6 

I. 
2( 

2; 

3< 

3-S 

D.C 
7.3 

M 

3 

27 
28 
29 

949958 
949996 
9.50034 
9.  50  072 
9.50  no 

3° 

38 

£ 

38 

o9 

9,52241 
9.52  284 
9.52326 
9.52368 
9.52410 

42 
42 
42 
42 
42 

0-47  758 
0.47  716 
0.47  674 
0.47  632 
0.47  590 

9*97716 
9.97712 

9-97  7o§ 
9.97  704 
9.97700 

4 
4 
4 
4 
4 

35 

34 
33 
32 
3i 

6 

7 
8 

3< 

3 

4 
5 

) 

9 

2 

3 

3 
4 

5 

8 
•§ 

38 

3-8 
4-4 
5-o 

30 

3i 
32 
33 
34 

9.5014? 
9.50185 
9.50223 
9.  50  260 
9.50293 

37 
38 

I 

_a 

9.52452 
9.52494 
9.52536 
9.52578 
9.52619 

42 
42 
42 
42 

4? 

0-47  548 
0.47  506 
0.47  464 
0.47  422 
0.47  386 

9.97695 
9-97  691 
9-97  687 
9.97  683 
9.97673 

4 
4 
4 
4 
4 

30 

29 
28 
27 
26 

9 

10 

20 
30 
40 

1 

13 
19 

26 

cS 

S 

0 

5 

0 

6 

12 

'9 

25 

.8 

'1 

.2 

•6 

5-7 
6-3 

12.6 

19.0 
25.3 

!i 

37 
38 
3Q 

9-50336 
9.50373 
9.50411 
9.50443 
9.  $0  486 

37 

I 

3? 

Oa 

9.52  66  1 
9.52703 

9.52745 
9.52787 
9.52828 

42 
42 
41 
42 

4t 

.  y 

o.47  338 
0.47  295 
0.47  255 
0.47  213 
0.47  171 

9.97674 
9.97  670 
9.97  666 
9.97661 

9.9765? 

4 
4 
4 
4 
4 

25 
24 

23 

22 
21 

5° 

6 
7 

32 

3 
3 

5 

7 

7 
i 

32 

3 

3 

•A 

7 

•7 

9 

3]-6 

36 

3.6 

A   2 

40 

4i 
42 
43 
44 

9.50523 
9.50561 
9.50598 
9.50635 
9.50672 

37 
37 

5 

37 
•»4 

9.52870 
9.52912 

9-52953 
9.52995 

9-53036 

4i 
42 
4? 
4t 
4? 

0.47  130 
0.47  088 
0.47  046 
0.47  005 
0.46  963 

9-97653 
9.97649 
9-97  644 
9-97  646 
9.97636 

4 
4 
4 
4 
4 

20 

19 
18 

17 
16 

8 
9 

10 

20 

3° 

S 

1 

12 

18 

0 

6 

2 
5 

A 

6 
12 

jS 

-9 

ij 

•  5 

4.8 

5-5 
6.1 

I2.I 

1  8.2 

45 
46 

47 
48 

49 

9.50710 

9-50747 
9.50783 
9.50821 
9.50853 

37 
37 
3? 
37 
37 

9.53078 

9.53II9 
9.53161 
9.53202 

9-53244 

41 
41 
4* 
4f 

4? 

0.46  922 
0.46  886 
0.46  839 
0.4679? 
0.46  756 

9.97632 
9.9762? 
9.97623 
9-976i9 
9.97614 

4 
4 
4 
4 
4 

15 
H 
13 

12 
II 

40 
So 

o 
31 

0 
2 

24 

3° 

4 

•6 

•8 

4 

24-3 
30.4 

50 

5i 
52 
53 
54 

9.50895 
9.50932 
9.50969 
9.51005 
9.51  043 

3> 
37 
37 
37 
37 

9.53285 

9.53326 
9-53368 

9.53409 
9-53450 

4i 
4i 
41 
4i 
4? 

0.46714 
0.46  673 
0.46  632 
0.46  591 
0.46  549 

9.97  616 
9.97606 
9.97601 
9-97  59? 
9-97593 

4 
4 
4 
4 
4 

10 

6 

0 

8 

9 

10 

C 

C 

c 
c 
c 

).4 

;ii 

>-7 

>.? 

0 

o. 

0. 
0. 
0. 

4 
4 

6 

P 
? 

59 

9.  5  1  086 
9.51  117 

9.51  154 
9.51  196 

9.51  22? 

37 
36 
37 
36 
37 

9.53491 
9-53533 
9-53574 
9.53615 
9-53656 

41 
4i 
4i 
4i 
4i 

0.46  508 
0.46467 
0.46426 
0.46  385 
0.46  344 

9-97  588 
9-97  584 
9-97  580 
9-97  575 
9.97  571 

4 

4 

4 

4 
4 

5 
4 
3 

2 
I 

30 
40 

50 

2 

,2 

.0 

-7 

2. 
2. 

3- 

0 

00 

9.51  264 

36 

9-53697 

41 

0.46  303 

9-97  567 

4 

0 

-- 

Los.'.  Cos. 

d. 

Los:.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

f 

!• 

I*. 

71 


366 


TABLE  VII.— LOGARITHMIC  SIXES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

19° 


/ 

Los.  sin. 

•1. 

I.-.-.  Tail.  ! 

c.d.i 

Log.  Cot. 

Log.  <  u». 

d. 

p. 

P. 

0 

^ 

3 
4 

9.51  264 
9.51  3OI 
9-51  337 
9-51  3/4 
9.51  410 

37 
36 
36 
36 

9.53697 
9-53738 
9-53779 
9-53820 
9.53861 

41 

41 
41 
41 
AI 

0.46  303 
0.46  262 
0*46  221 
0.46  1  80 
0.46  139 

9-97  567 
9-97  562 
9-97  558 
9-97  554 
9-97  549 

4 
4 
4 
4 
2_ 

60 

59 
58 

% 

6 

4i 

4-i 

46 

4.6 

40 

4.0 

6 

7 
8 

9 

9-5»  447 
9.51483 
9.51  520 
9-5i  556 
9-5i  593 

36 
36 
36 

9-53902 
9-53943 
9-53983 
9-  54  024 
9.54065 

41 

46 

41 
41 

AO 

0.46098 
0.46057 
0.46015 

0.45  975 
0-45  934 

9-97  545 
9-97  54i 
9-97  536 
9-97  532 
9-97  527 

4 
4 
4 
4 

55 
54 
53 
52 
5i 

7 
8 

9 

10 
20 

4.8 
5-4 
6.1 

6-8 
13-6 

4-7 

« 

6-7 
13-5 

4-6 

K 

6-6    j 
13-3 

10 

ii 

12 
13 

U 

9.51  629 
9.51665 
9.51  702 
9-5I738 
9-5I774 

| 

36 
36 

36 
7< 

9.54106 

9-54I47 
9.54187 
9.54228 
9-  54  269 

41 
40 
46 

41 

AO 

0.45  894 

0-45  853 
0.45  812 
0.45  772 
0-45  731 

9-97  523 
9-97  519 
9-97  5H 
9-97  510 
9-97  5o5 

4 
I 

4 
4 

50 

49 
48 
47 
46 

3° 
40 

50 

20.5 
27-3 
34-1 

3 

27.0 
33-  f 

9      2 

26.5 
33-3 

9 

15 

16 

17 
18 

19 

9.51  816 
9.51  847 
9.51883 
9.51  919 
9-5i  955 

3° 
36 

3^ 

^ 
36 

?f> 

9-54309 
9-54350 
9-54390 
9-54431 
9.54471 

4<-» 
40 
46 
40 
46 
AO 

0.45  696 
0.45  650 
0.45  609 
0.45  569 
o-45  528 

9.97501 

9-97497 
9.97492 
9-97488 
9-97483 

4 
4 
4 
4 
4 

45 
44 
43 
42 

4i 

6     3 
7     4 
8      5 
9     5 
10     6 

•9     2 
-6     A 
-2     5 

•9     5 

.6     t 

•9 

-5 

.2 

•8 
.5 

20 

21 
22 

23 
24 

9.51  991 
9.52027 
9.52063 
9.52099 
9-52  135 

3° 

^ 
36 

36 
36 

•28 

9.54512 
9-54552 
9-54  593 
9-54633 
9-54673 

46 
46 
40 
46 
4.6 

0.45  488 
0.45  447 
0.45  407 
0.45  367 
0-45  326 

9-97479 
9-97475 
9.97470 
9.97466 
9-9746i 

4 
4 

4 

4 
4 

40 

39 
38 

i 

i 
' 

20    13 

30    19 
j.o   26 
50   32 

i  I3 
•7  is 

•  3    26 
-9   32 

.0 

•5 

.0 

•5 

25 
26 

27 
28 

29 

9.52  170 
9.52205 
9.52242 
9.52278 
9-52314 

35 

$ 
t 

_  a 

9.547H 
9-54754 
9-54794 
9  54834 
9-  54  874 

40 
46 
40 
40 

An 

0.45  286 
0.45  246 
0.45  205 
0.45  16$ 
0.45  125 

9-97457 
9.97452 
9.97448 
9-97443 
9-97439 

4 
4 
4 

35 
34 
33 
32 
31 

6 
'   8 

37 

3-7 
4-3 

4"? 

36 

3-6 
4-2 
4.8 

36 

3-6 

4-2   i 
4-8   i 

30 

3i 
32 
33 
34 

9-52349 
9-52  385 
9.52421 

9-52456 
9.52492 

3!) 
35 
36 
35 
35 

0  ? 

9-549!5 
9-54955 
9-54995 
9-55035 
9-55075 

4U 
40 
40 
40 
40 
4.6 

0.45  085 
0.45045 
0.45  005 
0.44965 
0.44925 

9-97434 
9-97  43° 
9-97425 
9.97421 

9-97416 

4 
4 
4 
4 
4 

30 

29 
28 
27 
26 

9 

10 
20 
30 
40 
CO 

5-5 
6.1 

12.3 
18.5 

24.5 

•JQ    Q 

5-5 

6.1 

12.  1 

18.2 

24.3 

•JQ  A 

5-4 
6.0 

12.0 

18.0 
24.0 

•3Q   n      ! 

35 
36 

3 

39 

9.52527 
9.52563 

9-52598 
9.52634 
9.52669 

3i> 
35 

?! 

35 

9.55H5 
9-55I55 
9-55I95 
9-55235 
9-55275 

39 
40 

40 
4o 

AO 

0.44884 
0.44  845 
0.44  805 
0.44  765 
0.44725 

9.97412 
9-9740? 
9-97403 
9-97  398 
9-97  394 

4 
4 
4 
4 
4 

25 
24 
23 

22 
21 

J^ 

6 
7 

J^-O 

35 

I! 

3r**+ 

35 

3.5 
4.1 

34 

3-4 
4.0 

40 

4i 
42 
43 
44 

9.52704 
9.52740 

9'>217> 
9.52  810 

9.52846 

35 

11 

35 
35 

9-553I5 
9-55355 
9-55394 
9.55434 
9.55474 

40 

39 
40 

39 

0.44685 
0.44645 
0.4460^ 
0.44  565 
0.44  526 

9-97  389 
9-97  385 
9-97  380 
9.97376 
9-97371 

4 
4 
4 
4 
4 

20 

J9 
18 

17 
16 

8 
9 

10 
20 
30 

4-? 
5-3 

5-? 
n.  8 
17.? 

4-6 

r.i 

n-6 
17-5 

4-6 

% 

11.5 
17.2 

41 
46 

47 
48 
49 

9.52881 
9.52916 
9.52951 
9.52985 
9.53021 

35 

35 
3! 
35 
35 

9-555H 
9-55553 
9-55593 
9.55633 
9.55672 

40 

39 
40 

39 

39 

-A 

0.44  486 
0.44445 
0.44405 

0.44367 
0.44327 

9-97367 
9-97  362 
9-97  35s 
9-97  353 
9-97349 

4 
4 
4 
4 
4 

15 
14 
13 

12 
II 

40 
50 

23-6 
29.6 

5 

S? 

4 

23.0  | 
28.? 

4 

50 

5i 
52 

53 
54 

9-53056 
9.53091 

9-53I26 
9.53161 

9-53196 

35 
35 
35 
35 
35 

1  A 

9.55712 
9-55751 
9-55791 
9-55831 
9.55876 

39 

39 
40 

39 
39 

0.44  288 
0.44  248 
0.44  203 
0.44  169 
0.44  129 

9-97  344 
9-97  340 
9-97  335 
9-97  330 
9-97326 

4 
4 
5 
4 
4 

10 

I 

6 

< 

. 

I 

5  0.5 
7   c.6 
8   0.5 
9  °-7 
oo.§ 

0.4 
°1 

0.6 
0.7 
o.f 

0.4 

0.4 

°i 

0.6 
0.5 

P 
% 

59 

9-5323I 
9.53266 
9-53301 
9-53335 
9-53370 

34 
35 
35 
34 
35 

9.55909 
9-55949 
9-55988 
9.56028 
9.  56  06? 

39 
39 
39 
39 
39 

0.44096 
0.44051 
0.44011 
0.43  972 
o.43  932 

9-97  321 
9-97317 
9.97312 
9.97308 
9-97303 

4 

4 

* 

4 

5 

5 
4 
3 

2 

3 

4 
5 

1-o 

o   2.5 

3   3-3 

D    4.1 

l\ 

2.0 

2.6 

3-3 

60 

9-53405 

34 

9.56105 

39 

0.43893 

997298 

4 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

P 

p. 

TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

20° 


, 

Log.  hin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p. 

p 

0 

I 

2 

3 
4 

9.53405 
9-53440 
9-53474 
9.53509 
9-53544 

35 
34 
34 
35 
•zl 

9o6  105 
9.56  146 
9.56185 
9.56224 
9.56263 

39 
39 
39 

39 

0.43  893 
0.43  854 

0.43815 
0-43775 
0.43736 

9-97  293 
9-97  294 
9.97  289 
9.97  285 
9.97  280 

4 
4 
4 

5 

4~ 

60 

59 

1 

6 

3 
3 

9 

•  9 

3 
3 

9 

-9 

7 
8 

9 

9-53578 
9-536I3 
9.5364? 
9.53682 

9-53716 

34 
34 
34 
34 
34 

9.56303 
9.56342 
9.56381 
9.  56  420 

39 
39 
39 
39 
39 

0.43  697 
0.43658 
0.43619 
0.43  580 
0-43  540 

9-97  275 
9.97271 
9-97  265 
9.97261 
9.97257 

4 

4 

55 
54 
53 
52 

8 
9 

10 
20 

4 

5 

6 
13 

.6 

.2 

•9 
.6 
.1 

4 
5 

6 
13 

•5 

.2 

-8 
•5 
.0 

10 

ii 

12 
13 
14 

9-53750 
9-53785 
9-538I9 

9.53888 

34 

34 
34 

34 
34 

0? 

9.56493 

9.56576 
9.56615 
9.56654 

39 
39 
39 
39 
38 

0.43  501 
0.43462 
0-43  423 
o.43  384 
0-43  346 

9.97252 
9.97  248 
9-97  243 
9-97238 
9-97  234 

4 

4 

4 
4 

50 

49 
48 

47 
46 

3° 
40 

3 

26 
32 

g 

./ 
•3 
•9 

3 

11J 

2r 

32 

8 

•5 

.0 

•5 
3? 

:i 

17 

18 

19 

9.53922 
9-53956 
9-53990 
9.54025 
9-54059 

34 
34 
34 
34 
34 

9.56693 

9.56771 
9.56  810 
9.56843 

39 
39 
39 
38 

0-43  307 
0.43  268 
0.43  229 
0.43  190 

0.43  151 

9.97  229 
9-97  224 
9.97  220 
9.97215 
9.97216 

I 
4 

45 
44 
43 

42 

6     3 
7     4 
8     5 
9     5 
10     6 

8 
5 

8 
-1 

3 
4 
5 

I 

.8 

•4 

.6 

-7 
.  3 

3-? 
4-4 
5-° 
5-6 

6.2 

20 

21 
22 
23 

!         24 

9.54093 
9.54127 
9.54161 

9  54195 
9.54229 

34 
34 
34 
34 
34 

9.56887 
9.  56  926 

9-56965 
9.57003 
9.57042 

39 
38 
39 
38 
38 

0.43  112 
0.43074 
0.43035 

0.42  995 
0.42  958 

9.97  206 

9.97  201 

9-97  196 
9-97  191 
9.97  187 

4 

5 

4 
o 

40 

39 
38 
37 
36 

20    12 
30    19 

40    25 
50    32 

8 
•6 

12 

25 
31 

•  6 

.0 

'1 

12.5 

18.? 
25.0 
31.2 

25 
26 
27 
28 
29 

9.54263 
9-54297 
9-54331 
9.54365 
9-  54  398 

53 

34 
34 
34 
33 

9.57081 
9.57119 
9.57158 

9-57  196 
9-57235 

39 

38 
38 
38 
38 

0.42  919 
0.42  886 
0.42  842 
0.42  803 
0.42  765 

9.97  182 
9.97  17? 
9-97  173 
9.97  168 
9-97  163 

4 

5 
4 

4 

35 
34 
33 

32 

3 

6     3 
7     A 
8     A 

5 

•5 
.  i 

•6 

9 

2 

i 
t 

4 

3-4 
pc 

J..6 

34 

3-4 
3-9 

4-5 

80 

32 
33 
'    34 

9-  54  432 
9.54465 
9.54500 
9-54534 

34 
34 
33 
34 
33 

9.57274 
9.57312 
9-57350 

9-5742? 

39 
38 
38 
38 
38 

0.42  726 
0.42  68? 
0.42  649 
0.42611 

0.42  572 

9.97159 
9.97154 
9.97  149 
9-97  144 
9-97  HO 

4 

4 

30 

29 
28 

27 
26 

9     5 
10     5 

20     II 

30     17 
40    23 
CQ     2Q 

.-1 

-8 
•6 
•5 
-3 

i 

i; 

2' 
2i 

r? 

•5 

7.2 

h° 

5-6 
n-3 
17.0 

2  8*.  5 

35 
36 

11 

39 

9.54601 

9-  54  663 
9-  54  702 
9-54735 

33 
33 
34 
33 
33 

9.57466 
9.57504 
9-57  542 

9.57619 

38 
38 
38 
38 
38 

0.42  534 
0.42  495 
0.42  457 
0.42419 
0.42  386 

9-97  135 
9.97  136 

9-97  125 
9.97  121 
9.97  116 

4 
5 

25 
24 

23 

22 
21 

6 
7 

3 

3 
\ 

3 
•§ 

.0 

3 

3 
3 

3 

-3 

•  8 

40 

42 

43 

1    44 

9-  54  769 
9.54802 
9.54836 
9-54869 
9  54  902 

33 

II 

33 
33 

9-5765? 
9.57696 

9.57734 
9-57772 
9.57816 

38 
3? 

38 

0.42  342 
0.42  304 
0.42  266 

0.42  22? 
0.42  l8§ 

9.97111 

9-97  105 
9.97  102 
9.97097 
9-97092 

4 

4 
5 
5 

20 

19 
18 

17 
16 

8 
9 

10 
20 
30 

4 
5 

5 

i  r 

•4 
.0 
.6 
.T 
•7 

4 
4 

5 

1  1 
1  6 

•4 
•9 

•5 

.0 

-5 

4| 
46 

47 
48 
49 

9-54936 
9-  54  969 
9.55002 
9.55036 
9.55069 

33 
33 
3§ 
33 

9-57848 
9.57886 

9.57925 
9.58001 

3d 
38 
38 
38 
38 

o 

0.42  151 
0.42  113 
0.42075 
0.42  037 

0.41  999 

9.97087 
9.97082 
9.97078 
9.97073 
9.97063 

4 

5 
4 
5 

4 

15 
14 
13 

12 
II 

40 
50 

22 

2/ 

•3 

•9 

5 

22 
27 

4 

.0 

•5 

50 

52 
53 
54 

9.55102 
9-55I35 
9-55  l68 
9.55202 

9-55235 

33 
33 

33 
33 

9.58039 
9.58077 
9.58115 

9.58153 
9.58196 

38 

38 
38 

if 

0.41  961 
0.41  923 
0.41  885 
0.41  847 
0.41  809 

9.97063 

9-97058 
9.97054 
9.97049 
9.97044 

5 
5 
4 

5 
5 

10 

9 
8 

7 
6 

6 

8 
9 

10 

0 
O 
0 

o 
o 

4 

0.- 
0. 

0. 
0. 
0. 

I 

5 
5 

7 
} 

1 

59 

9.55268 
9-55301 
9-55334 

9.55400 

33 
33 
33 
33 
33 

9.58223 
9.58265 
9.58304 
9.58342 
9.58380 

3^ 

I 

0.41  771 

0.41  733 
0.41  695 
0.41  658 
0.41  620 

9.97039 
9.97034 
9.97029 
9-97025 

9-97  020 

4 
5 

I 

5 

5 
4 
3 

2 
I 

3° 

40 

5° 

2 

3 
4 

•0 

•5 
•3 
.  i 

2. 

3-< 
3- 

I 

3 

60 

9-55433 

33 

9.5841? 

J/ 

0.41  582 

9.97015 

5 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

' 

p. 

p. 

-         i 

368 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

21° 


Log.  Siii. 

d. 

Log.  Tan.    r.  d.      Log.  Cot. 

Log.  Cos. 

d. 

p.  p. 

0 

I 

2 

3 

4 

9-55433 
9.55466 

9-55498 
9-55531 
9.55564 

33 
32 
33 
33 
32 
33 
32 
33 
32 
32 
32 
33 

II 

32 
32 
32 

II 
II 

32 
32 
32 

-a 

9.5841? 
9.58455 

9.5853I 
9.58563 

II 
II 

37 

1 

37 
37 
3? 

I 

3? 
37 
37 

II 

37 
3? 
37 
3? 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
37 
38 
37 
37 

§ 

37 
37 
38 
37 
38 
37 
38 

38 
38 
38 

1 

38 
38 

$ 

i 

38 

0.41  582 
0.41  544 
0.41  507 
0.41  469 
0.41  431 

9.97015 
9.97016 
9.97005 
9.97006 
9.96995 

4 
5 
5 
5 
4 
5 
5 

4 
5 
5 
5 
4 
5 
5 
5 
5 
5 
4 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 
5 

5 
5 

5 
5 

5 

00 

59 
58 

6 

8 
9 

10 
20 
30 
40 
50 

6 

8 
9 

10 
20 
30 
40 

50 

I 

2 

3 
4 
5 

3 

3 
-4 
5 

I 

12 

IS 

25 

31 

6 

8 
9 

10 

20 

30 
40 

50 

_ 
- 

I 
I( 
2. 
2J 

6 

8 
9 

10 

20 

30 
40 

50 

6 

B 

9 

0 
0 
0 
0 
0 

8 
.8 

.4 
.6 

•7 
•  3 
•8 

.0 

.3 

•8 

3 

3 
4 
4 

I 

12 

18 
24 
30 

(3 

;-3 

'•4 
\-9 
5-5 

[.0 

5-5 
z.o 

3 
3 
3 
4 
4 
5 

1C 

15 

21 
26 

3 

0.5 
0.5 

0.? 

0.8 

2.* 

3-6 
4-6 

3' 

3 
4 

5 
5 

6 

12 

18 
25 

6 

•6 

.2 

•  8 
•5 
.1 
.1 

.2 

-3 

•4 

3 

3 
3 
4 
4 
5 

10 

16 

21 
27 

I 
.1 

-7 

.2 

•7 

•5 
-7 

.0 
.2 

0 

O 
0 
0 
0 

I 

2 

3 

4 

7 

7 

_l 

0 

6 

2 

5 

o 

2 

2 
*. 

A 

A 
i 

i: 

R 
2_ 
3^ 

2 
.2 

.8 
-3 
•9 

•  4 

'.12 

-6 
.1 

. 
. 

i< 
i 

2^ 

2 

5 
6 

8 

-6 
•5 

37 

3-7 
4-3    ! 
4-9 

11 
III 

24-8    l 
3°.  8 

6 

(.6 

h.2 

^8 
•4 

i.O 

;!o 

[.0 

32 

if 

4.2 

4.8 

h 

16.0 
21.3 

26.8 

JI 

!:? 

U 
5-t 
3-3 
5-5 

5-8 

4 
0.4 

0.7 

0.? 
2^2 

i 

7 
8 

9 

9-55597 
9.55630 
9.55662 
9.55695 
9.55728 

9.58606 
9.58644 
9.58681 
9.58719 

0.41  394 
0.41  356 
0.41  313 
0.41  281 
0.41  243 

9.96991 
9.96986 
9.^6981 
9.96976 
9.96971 

55 

54 
53 
52 

10 

ii 

12 

13 
14 

9.55766 

9-55793 
9.55826 

9.55891 

9.58794 
9.58831 
9.58869 
9.58906 

9  58  944 

0.41  206 
0.41  i6§ 
0.41  131 
0.41  093 
0.41  056 

9.96965 
9.96961 
9-96958 

9.96947 

50 

49 
48 
47 
46 

15 

16 

17 
18 

19 

9.55923 
9.55956 
9.55988 

9.  56  020 
9-56053 

9.58981 
9.59019 
9.59056 
9.59093 
9-59I3I 
9.59168 
9.59205 
9.59242 
9.59280 
9-593I7 

0.41  013 
0.40981 
0.40944 
0.40905 
0.40  869 

9-96942 
9-96937 
9.96932 
9.96927 
9.96922 

45 
44 
43 
42 

~~!o~ 

39 
38 
37 
36 

20 

21 
22 

23 

j      24 

9.56085 
9.56118 
9.56150 
9.56l82 
9.56214 

0.40  832 
0.40  794 

0.40757 
0.40  720 
0.40683 

9.9691? 
9.96912 

9-96907 
9.96902 
9.9689? 

27 
28 
29 

9.56247 
9.56279 
9.56311 
9.56343 

32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 
32 

32 
32 
3? 

32 
3* 

32 

3? 
3i 
3i 

3i 
3t 

l\ 

3i 

9-59354 
9-59391 
9.59428 
9-  59  465 
9.59502 

0.40  646 
0.40  6o§ 
0.40  571 
0.40  534 
0.40  49? 

9.96892 
9.9688? 
9.96882 
9.9687? 
9.96873 

35 
34 
33 
32 

30 

32 
33 
34 

9.5640? 

9-  56  439 
9.56471 

9-56503 

9-  59  540 
9-59577 
9.596i4 
9.59651 
9.  59  688 

0.40  460 

0.40423 
0.40  386 
0.40  349 
0.40  312 

9.96868 
9-96863 
9.96858 

9^96848 

30 

29 
28 
27 
26 

35 
36 

fs 

39 

9.56567 
9.56599 
9.56631 
9.56663 
9-  56  695 

9o9724 
9.5976T 
9-59798 

9.59872 

0.40275 
0.40  238 

0.40  201 

0.40  164 
0.40  128 

9.96  843 
9.96838 

9.96828 
9.96823 

25 
24 
23 

22 
21 

40 

42 
43 

44 

9.56727 
9-56758 
9-56790 
9.56822 
9.56854 

9.59909 
9.59946 
9-59982 
9.60019 
9  60056 

0.40091 
0.40054 
0.4001? 
o.  39  986 
0-39944 

9.96818 
9.96813 
9.96808 
9.96802 
9-96797 

20 

19 
18 

17 
16 

4I 
46 

47 
48 

49 

9.56885 
9.56917 
9.56949 
9.  56  986 
9.57012 

9.60093 
9.60  129 
9.60  165 
9.60  203 
9.60239 

0.39907 
0.39876 

0.39833 
0-39797 
0.39766 

9.96792 

9.96  78? 
9.96782 

9-96  772 

15 
H 
13 

12 
II 

50 

52 
53 
54 

9-57043 
9.57075 

9-57106 
9.57138 
9.57169 

9.60276 
9.60  312 
9.60  349 
9.60  386 
9.60422 

0.39724 
0.39687 
0.39656 
0.39614 

0-3957? 

9.9676? 
9.96762 

9.96752 
9.96747 

10 

6 

:  | 

59 

9.57201 
9.57232 
9.57263 
9-57295 
9-57326 

9.60459 
9.60495 
9-60531 
9.60  568 
9.60604 

0-39541 
0.39504 

0.39468 
0.39432 
0-39395 

9.96742 

9.96732 
9.96727 
9.96721 

5 
4 
3 

2 

I 

60 

9-57  35? 

31 

9.60641 

0-39359 

9.96715 

0 

Loe.  Cos.        d. 

Loe.  Cot.     c.  d.      Log.  Tan. 

Log.  Sin.         d. 

'                  p.  p.              1 

68( 


369- 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

22° 


r 

Log.  Siii. 

d. 

Log.  Tan.  |  c.  d. 

Log.  Cot. 

Log.  Cos.   i      d. 

P.  P. 

0 

I 

2 

3 

4 

9-5735? 
9-57389 
9.  57  420 

9-57451 
9.57482 

31 
31 

11 

31 
31 
3? 
31 
31 
31 
31 
31 
31 
35 
31 
31 
31 
30 
31 

-20 

9.60  641 
9.6067? 
9.60713 
9.60750 
9.60786 

36 
36 
36 
36 
36 
36 

% 
36 

36 

^ 
36 

36 

% 
36 

36 

% 
36 

% 
36 

36 
36 
36 
36 
35 
36 
36 

I 

36 

11 

35 
35 

36 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

1! 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 

0.39359 
0.39322 

0.39286 
0.39250 
0.39213 

9.96716 
9.96711 
9.96706 
9.96701 
9.96696 

5 

5 

5 
5 
5 

5 
5 
5 
5 

5 
5 

5 
5 
5 
5 
5 

5 

5 
5 

5 
5 
5 

5 

5 
5 

5 
5 

"    5 

5 
5 
5 

5 
5 

5 
5 

5 
5 

60 

59 
58 
57 
56 

6 

8 

9 
10 

20 

30 
40 

5° 
6 

I 

9 

10 
20 
30 
40 

50 

6 

I 

9 

10 
20 

30 
40 
50 

•: 

6     } 
7     2 
8     A 

9     A 
10     5 

20    1C 

30    15 
40    2C 

50    25 

( 

< 
K 
2( 

3< 
4< 
5< 

36 

3-6 
4.2 

4-8 
5-5 
6.1 

I2.T 

18.2 

24.3 

30.4 

35 

3-5 
4-i 

4-? 
5-3 
5-9 
ii.  8 
17.? 

23-6 
29.6 

31 

3-1 

3-7 
4.2 

4-7 
5.2 
10.5 
I5-? 

21.0 
26.2 

6      3 

-o     3 

•I     3 
..o     4 

.6     4 

-i     5 
.1    10 

.2     I5 

>-3   20 
•4  25 

$ 

>  o.§ 

'   o.g 
5   o.? 
)  0.8 
5  0.9 

>    1.8 

>    2.? 

>   3-6 
>  4.6 

36 

3.6 

4.2 
4.8 

5-4 
6.0 

12.0 

18.0 
24.0 
30.0 

35 

3-5 
4.1 

4-6 
5-2 
5-8 
ii.6 
17-5 
23-3 
29.1 

3i 

3-i 
3-6 
4-1 
4-6 
5-1 
10.3 
15-5 

20.g 

25-8 

o      29 

.0     2.9 

.5  3.4 

-o     3-9 
•5     4-4 
.0     4.9 

.0       9.§ 

.0   14.? 
.0   19.^ 
.0  24.6 

5 
0.5 
0.6 
0.6 

0.? 

o.§ 
1-8 

2-5 

i? 

6 

8 
9 

9-57  5'3 
9-57544 
9-57576 
9.57607 
9.57638 

9.60822 
9.60859 
9.60895 
9.60931 
9.6096? 
9.61  003 
9.61  039 
9.61  076 
9.61  112 
9.61  148 

0-39  177 
0.39  141 
0.39  105 
0.39069 
0.39032 

9.96691 
9.96686 
9.96681 
9.96675 
9.96676 

55 
54 
53 
52 
5i 

10 

ii 

12 
13 
-     H 

9.57669 
9.57700 

9-57731 
9.57762 
9.57792 

0.38996 

o.  38  966 

0.38924 
0.38888 
0.38852 

9.9666$ 
9.  96  660 
9.96655 
9.96650 
9-96644 

50 

49 
48 

47 
46 

15 

16 

17 
18 

19 

9.57823 

9-57854 
9.5788$ 
9.57916 
9-57947 

9.6l  184 
9.6l  220 
9.6l  256 
9.6l  292 
9.6l  328 

0.38816 
0.38  780 
0.38744 
0.38  708 

0.38  672 

9.96639 
9.96634 
9.96629 
9.96624 
9.96  619 

45 
44 
43 
42 

4i 

20 

21 

22 

23 

24 

9-57977 
9-58  oos 
9.58039 
9.  58  070 
9.58  loo 

31 

30 
31 
30 

30 
31 
30 
30 
30 
30 
30 
33 

3° 
33 
30 
30 
30 
30 
30 

3n 
30 
30 
33 
30 

30 
30 
3o 
30 
30 
30 
30 
30 
30 
30 
29 
30 

30 
29 

30 
30 

9-  6  1  364 
9.61  400 
9.61  436 
9.61  472 
9.61  50? 

0.38636 

o.  38  600 
0.38  564 

0.38  528 

o.  38  492 

9-96613 
9.96603 
9.96603 
9.96598 

9-96  593 

40 

39 
38 
37 
3<? 

3 

27 
28 
29 

9.58131 
9.58  162 

9.58192 
9.58223 
9.58253 

9-6i  543 
9.61  579 
9.61  615 
9.61  651 
9.61  685 

0.38456 

0.38  426 

0.38385 
0.38349 
0.38313 

9.9658? 
9.96  582 
9.96577 
9-96  572 
9.96  567 

35 

34 
33 
32 
3i 

30 

3i 
32 
33 

34 

9.58284 
9-58314 
9-58345 
9-58375 

9.58  406 

9.61  722 
9.61  758 
9.61  794 
9.61  829 
9.61  865 

0.3827? 

0.38  242 
0.38  206 
0.38  176 

0.38  135 

9.96  561 

9-96  556 
9.96551 
9.96546 
9.96  546 

30 

29 
28 
27 
26 

35 
36 
37 
38 
39 

9.58436 
9.58465 
9.58497 
9.58527 
9-5855? 

9.61  901 
9-6i  936 
9.61  972 
9.62007 
9.62043 

o.  38  099 

0.38063 
0.38028 
0.37  992 

0-37  957 

9.96535 
9.96530 
9.96  525 
9.96519 
9.96514 

25 
24 
23 

22 
21 

40 

4i 
42 

43 

44 

9.58587 
9.58618 
9.58648 
9.58678 
9.58708 

9.62  078 
9.62  114 
9.62  149 
9.62  185 

9.62  220 

0.37921 
0.37  886 
0.37  856 
0.37815 
0.37779 

9.96  509 
9.96  503 
9.96493 
9.96493 
9.96488 

20 

19 
18 

17 
16 

45 
46 
47 
48 

49 

9-58738 
9.58769 

9.58799 
9.58829 

9.58859 

9.62  256 
9.62  291 
9-62  327 
9.62  362 

9-62  397 

0-37  744 
o.37  708 
0.37673 
0.3763? 
0.37  602 

9.96482 
9.9647? 
9.96472 

9.96466 
9.96461 

15 
H 
13 

12 
II 

50 

5i 
52 
53 

54 

9.58889 
9.58919 

9-58  949 
9.58979 
9.59009 

9-62  433 
9.62  463 
9-62  503 
9.62  539 
9.62  574 

0.37  567 
0.37  53i 
0.37496 
0.37461 
0.37426 

9.96456 
9-96450 
9-96445 
9.96440 
9.96434 

10 

6 

55 
56 

% 

59 

9-59038 
9-59068 

9-59098 
9.59128 
9.59158 

9.62  609 
9.62  644: 
9.62  679 
9.62715 
9.62750 

0.37390 

0-37355 
0.37  326 
0.37285 
0.37250 

9.96429 
9.96424 
9.96413 
9.96413 
9.  96  408 

5 

4 
3 

2 

GO 

9.59  188 

9.62785 

0.37215 

9.96  402 

0 

Log.  Cos.        (1. 

Log.  Cot.     c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

p.  P.                | 

17O 


TABLE  VII. —LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

23° 


/ 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos.  1      d. 

p.  p. 

0 

I 

2 

3 
4 

9.59  1  88 
9.59217 
9.59247 
9.59277 
9-59306 

29 

3° 

29 
2§ 

30 

29 
2§ 

29 
2§ 

2§ 
29 
29 
29 
2§ 

29 
2§ 

29 
29 

29 

29 
29 

29 
29 

29 

29 
29 
29 

29 
29 

29 

29 
29 

29 
29 

29 
29 
29 
28 

29 
29 

28 

29 

28 

29 

28 

29 

28 

28 

29 

28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 

9.62  785 
9.62  826 

9.62855 

9.62  896 

9.62  92  § 

35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
35 
34 
35 

3 

35 

11 

35 
34 
34 

3| 

34 

34 
35 
34 
33 
34 
34* 
3* 
34 

11 

A 
$ 

34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 
34 

34 
34 
34 
34 

0.37215 
0.37  179 
^0-37  144 
0.37  109 
0.37074 

9.96402 
9.96397 
9.96392 
9.96386 
9.96381 

5 

5 

5 
5 
5 

5 
5 

5 

5 
5 

5 
5 

5 
5 

5 

5 
5 

5 

6 

5 

5 

5 
5 

6 

5 

1 

I 

5 

60 

11 
I 

6 

8 
9 

10 
20 
30 
40 

50 

6 

7 
8 

9 

10 
20 
30 
40 
50 

2 

6     : 

I  : 

9    < 

10      i 

20    < 
30  \i 
40  i< 

50    2; 

6 

7 
8 

9 

10 
20 
30 
40 
50 

3 

3 
4 
4 
5 
5 
ii 

17 
23 
29 

3 

5 
4 
4 

5 

5 
ii 

17 

23 
28 

6 

8 
9 

10 
20 
30 
40 
50 

>9 

2-9 

5-4 
5.9 
^4 
t-9 
?-8 
*•? 
?-6 
^6 

6 

0.6 
0.7 
0.8 
0.9 

I.O 
2.0 

3-o 
4.o 
5.0 

!> 
.j 

-f 

•3 

•9 
•S 

>, 

.6 

4 

•4 

.0 

.6 

.2 
•? 

.O 

•7 

3 
3 
2 
4 
4 
5 

1C 

15 

2C 

-5 

2 
2 

3 

5 
4 
4 
9 
14 
19 
24 

0. 

o. 

0. 

0. 

o. 
I. 

2. 

3- 

4- 

i 
i 

I 

i 
2 
2< 

i 
\ 

"3 

2: 
2i 

o 

.0 

.5 

.0 

•5 

.0 

.0 
.0 

.0 
0 

9 

•9 

•  4 

§ 

6 

I 

• 

} 

8 

1 
1 

J5 

5-5 
M 
t-6 

r.3 

f-6 
7-5 
3.3 
*l 

34 
J.4 
5.9 
tf 

;.i 
>.g 

t-3 

r.O 

;:| 

28 

2.8 
3-3 
3-8 
4.3 
4-? 

9-5 

14.2 

19.0 
23-? 

5 

°1 
0.6 

0.6 
o.f 

0.8 
1.6 
2.5 

4-"t 

6 

8 
9 

9-59336 
9.59366 

9-59395 
9.59425 

9-59454 

9.62  966 
9.62  995 
9.63036 
9.63065 

9.63  106 

0.37039 
0.37004 
o.  36  969 

0.36934 
o.  36  899 

9.96375 
9.96  370 
9.96365 

9-96359 
9-96354 

55 
54 
53 
52 
5i 

10 

ii 

1  2 
13 
14 

9-  59  484 
9-595I3 
9-59543 
9.59572 
9.  59  602 

9-63  135 
9.63  176 
9.63  205 
9.63  240 
9.63  275 

0.36864 
0.36  829 
0.36794 
0.36  760 
0.36725 

9-96349 
9-96343 
9.96338 
9-96332 
9-96327 

50 

49 

48 

47 
46 

15 

16 

17 
18 

19 

9-59631 
9.59661 
9.59696 
9.59719 
9-  59  749 

9.63310 

9-63  344 
9-  63379 

9.63414 

9-63  449 

o.  36  690 

0.36655 

0.36  620 

0.36585 
0.36551 

9.9632? 

9.963I6 
9.96311 
9.96305 
9-96300 

45 
44 
43 
42 

4i 

20 

21 
22 
23 
24 

9-59778 
9.59807 

9-59837 
9.59866 
9.59895 

9.63484 
9-63518 
9-63  553 
9.63  588 
9.$3  622 

0.36  516 

0.36481 
0.36447 

0.36412 

0.3637? 

9.96294 
9.96  289 
9.96283 
9.96  278 
9.96272 

40 

39 

38 
37 
36 

3 

27 
28 

29 

9.59924 

9-59953 
9.59982 
9.60012 
9.60041 

9-63657 
9.63692 

9-63726 
9.63  761 

9-6379§ 

0.36343 

o.  36  308 

0.36273 
0.36239 

o.  36  204 

9.96267 
9.96  26! 
9.96256 
9.96251 
9.96245 

35' 
34 
33 
32 
3i 

30 

3i 
32 
33 
34 

9.60070 
9.60099 
9.60  128 
9.60157 
9.60  186 

9.63  830 
9.63  864 
9-63899 
9.63933 
9.63968 

0.36  170 

0.36135 

0.36  ioi 
o.  36  065 
0.36  032 

9.96  240 
9.96234 
9.96229 
9.96  223 
9.96  218 

30 

29 
28 

27 
26 

P 

37 
38 
39 

9.60  215 
9.60244 
9.60273 
9.60301 
9.60330 

9.  64  002 
9.64037 
9.64071 

9.64  106 
9.64  146 

0.3599? 
0.35963 
0.35923 
0.35894 

0.35859 

9.96  212 

9.96  205 

9.96  201 
9.96  195 
9.96  190 

25 
24 

23 

22 
21 

40 

4i 

42 

43 
44 

9.60359 
9.60388 
9.60417 
9.  60  44  § 
9.60474 

9.64  174 
9.64  209 
9.64243 

9-6427? 
964312 

0.35825 
0.35791 

0-35756 
0-35  722 
0.35688 

9.96  184 
9.96  179 
9.96173 

9.96  1  68 
9.96  162 

20 

19 
18 

17 
16 

45 
46 

47 
48 

49 

9.60  503 
9.60  532 
9.60  566 
9.60  589 
9.60618 

9-64  346 
9.64386 
9.64415 
9.64449 
9.64483 

0-35653 
0.35619 

o.35  585 
0-35551 
o.355i7 

9.96157 
9.96151 
9.96  146 
9.96  140 
9-96  134 

15 
H 

13 

12 
II 

50 

5i 
52 
53 
54 

9.60645 
9.60  675 
9.60  703 
9.60732 
9.60760 

9.64517 
9.64551 
9.64585 
9.64620 
9.64654 

0.35482 

0-35448 
0.35414 
0.35380 
0.35346 

9.96129 
9.96  123 
9.96  118 
9.  96  1  1  2 
9-96  105 

10 

6 

P 

57 
58 
59 

9.60789 
9.6081? 
9.60846 
9.60874 
9.60903 

9.64688 
9.64  722 
9.64756 
9.64  796 
9.64824 

0.35312 
0.35278 

0.35244 
0.35209 

0.35175 

9.96  101 
9-96095 
9.96090 
9.96084 
9.96078 

5 

4 
3 

2 
I 

60 

9.60  931 

9-64858 

0-35  HI 

9-96073 

0 

Log.  Cos.        d.        Log.  Cot.     P.  d.      Lor.  Tan. 

Log.  Sin.    i      d. 

/ 

P.  P. 

fifi 


371 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 


1  ' 

Log.  Sin. 

(1. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

P.  P. 

? 

2 

3 

4 

9.60931 
9.60959 
9.60988 

9.61  oig 
9.61  044 

28 
28 

28 
28 

^0 

9.64858 
9.64892 
9.64926 
9.  64  960 
9.64994 

34 
34 
33 

34 

0.35  141 
0.35  10? 

0.35073 
0.35  040 

0.35  006 

9.96073 
9.  96  06? 
9.  96  062 
9.96056 
9.96056 

! 

5 

P 

(>0 

5| 
S8 

57 
56 

I      6 

8 
9 

9.61  073 
9.61  101 
9.61  129 
9.61  157 
9.61  1  86 

28 

28 

28 

28 

2§ 
~o 

9.65  028 
9.65  062 
9.65  096 
9.65  I2§ 
9.65  163 

34 
34 
34 
33 
34 

0.34972 
0.34938 
0.34904 

0.34876 

0.34836 

9.96045 
9.96039 
9.96033 
9.96028 
9.96022 

6 

1 

P 

55 
54 
53 
52 
5i 

3- 

6     3 
7     3 
8     4 
9     5 

I      3 

i    3 

9     3 

5     4 
i     5 

3      33 

•3     3-3 
.9     3-8 
•4     4-4 
.0     4.9 

10 

ii 

12 
13 

1      I4 

9.61  214 
9.61  242 
9.61  270 
9.61  293 
9.61  326 

28 
28 

28 

28 

•78 

9-65  19? 
9.65  231 
9.65  265 
9.65  299 
9.65  332 

35 

34 

3 

o.  34  802 
0.34769 
0.34735 
0.34701 
0.3466? 

9.96016 
9.96011 
9.96005 

9-95999 
9.95994 

! 

50 

49 
48 

47 
46 

10     5 

20    II 
30    17 
40    22 
50    28 

6     5 
3   ii 
o   16 

6    22 

3  27 

.6     5.5 

.1     II.  0 

.?'  16.5 

.§    22.0 
.9    27.5 

11 

17 

18 
19 

9-6i  354 
961  382 
9.61  416 
9-6i  438 
9.61  466 

28 
28 
28 
28 

08 

9-65  366 
9.65  400 

9-65  433 
9-6546? 
9.65  501 

34 
33 
33 

a 

•2J_ 

0.34633 
o.  34  600 

0.34566 
0.34532 
0.34499 

9.95988 
9.95982 

9-95977 
9.95971 

9-95  965 

6 

5 

45 

44 
43 
42 
4i 

28 

28 

20 

21 

22 

23 

i      24 

9.61  494 
9.61  522 
9.61  550 
9.61  578 
9.61  606 

28 

2? 

28 
28 

--,0 

9.65  535 
9.65  563 
9.65  602 
9-65635 
9.65  669 

jt 

33 
33 

1! 

•2/1 

0.34465 
o.3443i 
0.34398 
0.34364 
o.3433i 

9-95959 
9-95  954 
9.95  948 
9.95942 
9-95937 

I 

5 
f. 

40 

39 
38 
37 
36 

6 

7 
8 

9 

10 

2.8 
3-3 
3-8 
4-3 

4-? 

2.8 

3.2 
3.? 

4.2 

4-6 
r\  *? 

25 
26 
27 
28 
29 

9.61  634 
9.61  66i 
9.61  689 
9.61  717 
9.61  745 

2? 
28 
2? 
28 

•?9 

9.65  703 

9-65  73§ 
9.65  770 
9.65  803 
9-65  837 

j4 

33 
33 
33 
33 

•23 

0.34297 
0.34263 
0.34230 

0-34196 
0.34163 

9-95  93i 
9-95  925 
9.95919 
9.95914 
9-95  908 

6 

I 

P 

35 
34 
33 
32 
3i 

20 
30 
40 
50 

9-5 
14.2 
19.0 
23-? 

9-3 
14.0 

18.6 
23.3 

30 

3i 

11 

,34 

9.61  772 
9.61  806 
9.61  828 
9.61  856 
9.61  883 

27 
28 
2? 
28 
2? 
o9 

9.65  876 
9.65  904 

9-65  937 
9.65  971 
9.66004 

33 
33 
33 
33 
33 

•23 

0.34129 
0.34096 
o.  34  062 
o.  34  029 
0.33996 

9.95902 
9-95  896 
9-95  891 
9-95885 
9.95  879 

6 

6 
6 

P 

30 

29 
28 

27 
26 

6 

2? 

2.? 

27 

2.7 

1    35 
I    36 
37 
38 
39 

9.61  911 

9-6i  938 
9.61  966 
9.61  994 
9.62021 

27 
2? 
2? 

28 

2? 

~G 

9-6603? 
9.66071 
9.66  104 
9.66  13? 
9.66  171 

33 
33 
33 
33 
33 

1  3 

0.33962 
0.33929 

0.33895 
0.33862 
0.33829 

9.95873 
9.9586? 
9-95862 
9-95  856 
9-95  856 

i> 
6 

6 

5 

25 

24 
23 

22 
21 

8 
9 

10 
20 

3-2 

3-6 
4.1 
4.6 
9-t 

T  ?    5 

3'J 
3-6 

4.6 

4-5 
9.0 

40 

4i 
42 

43 

I    44 

9.62049 
9.62076 
9.62  104 
9.62  131 
9.62  158 

27 

2? 
2? 
2? 
27 

_0 

9.66  204 
9-66  23? 
9.66271 
9.66304 
9-6633? 

33 
33 
33 
33 
33 

0-33795 
0.33762 

0.33729 
0.33696 
0.33662 

9-95  844 
9-95  838 
9-95  833 
9.95  827 
9.95  821 

6 

i 

5 

20 

19 
18 

17 
16 

3° 

40 
50 

13-7 
l8-3 
22.9 

22.5 

45 
46 

47 
48 

49 

9.62  1  86 
9.62213 
9.62  241 
9.62  268 
9.62  295 

27 

2? 
2? 
27 

2? 

a 

9.66376 
9.66  404 
9.66437 
9.66476 
9-66  503 

33 
33 
33 
33 
33 

0.33629 
0-33  596 
0-33  563 
o.33  529 
0-33496 

9.95815 
9.95809 
9-95804 
9-95798 
9-95  792 

6 

6 
6 

15 
14 
13 

12 
II 

6 

6 

0.6 

0.7 

B 

o.S 

0.6 

50 

5i 

52 
53 
54 

9.62323 
9.62  350 

9-6237? 
9.62  404 
9.62432 

*l 

27 
2? 
27 
2? 

9-66  536 
9.66  570 
9.66  603 
9.66636 
9.66669 

33 
33 
33 
33 
33 

0.33463 
0.33430 
0-33397 
0.33  364 
0.33331 

9-95786 
9.95786 

9-95774 
9-95  76§ 
9-95  763 

5 

6 
6 
6 

5 

10 

9 
8 

6 

9 

10 
20 

30 
AO 

0.8 
0.9 

I.O 
2.0 

3-o 

40 

0.? 

0.8 
0.9 
1-8 

2.? 

•5    2 

55 
56 

11 

59 

9.62459 
9.62486 
9.62513 
9.62  546 
9.62  56? 

2? 

2? 
27 
27 

27 
a 

9.66  702 
9.66735 
9.66768 
9.66801 
9.66  83^ 

33 
33 
33 
33 
33 

0.33298 
0.33265 
0.33232 
0-33  198 
0.33  165 

9-95757 
9-95751 
9-95745 
9-95739 
9-95733 

6 
6 

6 

5 
4 
3 

2 

•P* 

50 

5.0 

4-6 

60 

9.62  595 

2/ 

9.66  86? 

33 

0.33  132 

9-95  72? 

6 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

p.p. 

TABLE  VII.— LOGARITHMIC  SIXES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

25° 


1 

Loe.  sin. 

d. 

IMS.  Tan. 

!o.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

P. 

p. 

0 

I 
2 

3 

4 

9.62  595 

9.62  622 
9.62  649 
9.62  676 
9.62703 

27 

27 
27 
27 

9.66  86? 
9.66900 

& 

9.66  999 

32 

33 
33 
33 

0-33  132 
0.33  ioo 

0:33067 
0.33034 

0.33001 

9.9572? 
9.95721 
9.95716 
9.95710 
9.95704 

6 

1 

6 
f. 

GO 

59 
58 
57 
56 

I 
4 

9 

9.62  730 

9.62  757 
9.62784 
9.62811 
9.62  838 

27 
27 

27 

27 

27 
o2 

9.67  032 
9.67065 
9.6709? 
9.67136 
9.67  163 

33 
33 
32 
33 
33 

0.32  968 

0.32935 

0.32  902 

0.32  869 
0.32835 

9.95  698 
9.95692 

9.95  686 
9.95686 
9.95674 

6 
6 

6 

55 
54 
53 
52 
5i 

6 

8 
9 

33 

% 

4.4 
4-9 

32 

*• 

4.3 
4.9 

32 

!i 

4.8 

10 

ii 

13 
13 
14 

9.62864 
9.62  891 
9.62913 
9.62  945 
9.62  972 

26 

27 
27 
27 
26 

9.67  195 
9.67  229 
9.67  262 

9  67  294 
9.6732? 

33 

32 
33 
32 
33 

_a 

0.32  803 
0.32771 
0.32  738 
0.32705 
0.32  672 

9-9566§ 
9.95662 

9.95656 
9.95656 
9.95644 

6 
6 
6 
6 

50 

49 
48 

47 
46 

10 
20 
30 
40 
50 

5-5 

II.  0 

16.5 

22.0 
27.5 

H 

10.8 
16.2 

21.6 
27.1 

,*I 

16.0 

21-3 
26.6 

15 

16 

17 

18 

19 

9.62999 
9.63025 
9.63052 
9.63079 
9.63  1  06 

27 

26 

27 

2£ 

27 
?z 

9.67  360 

9-67  393 
9.67425 

9-67458 
9.67  491 

32 

33 

32 
33 

32 
i3 

0.32  640 
0.32  607 
0.32  574 
0.32  541 
0.32  509 

9-95638 
9-95  632 
9-95  627 
9.95621 

9.95615 

6 

6 
6 

45 

44 
43 
42 
4i 

, 

27 

20 

21 

22 

23 

24 

9.63  132 
9.63159 
9.63  1  86 

9.63  212 

9.63239 

26 

27 

26 

26 
26 

9.67  523 

9-67  556 
9.67  589 
9.6762! 
9-67654 

32 
33 

i 

33 

_a 

0-32476 
0.32443 
0.32411 

0.32  378 

0.32345 

9.95609 
9.95  603 
9-95  597 
9-95  59i 
9-95  585 

6 
6 
6 
6 

40 

39 
38 

| 

6 

I 

9 

10 

2.7 

H 

4.6 

4-5 

3 

27 
28 

29 

9.63  266 
9.63  292 

9-63319 
9-63345 
9.63372 

27 

2§ 
26 

26 

26 
9? 

9.67  687 
9.67719 
9.67  752 

9-67784 
9.67817 

32 

II 
% 

-a 

0.32313 
0.32  286 
0.32  248 
0.32215 
0.32  183 

9-95  579 
9-95  573 
9-95  567 
9-95  561 
9-95555 

6 
6 
6 
6 

35 
34 
33 
32 
3i 

20 

.  30 
40 

50 

9.0 

I3o 
18.0 
22.5 

30 

3i 
32 
33 

34 

9-63398 
9.63425 

9-63451 
9.63478 
9.63  504 

26 

26 

26 

1 

26 

0? 

9.67  849 
9.67  882 
9.67  914 
9.67  947 
9.67979 

32 
32 

I 

0.32  150 
0.32  118 
0.32085 
0.32053 

0.32  020 

9-95  549 
9-95  543 
9-95  537 
9-95  530 
9-95  524 

6 

6 

30 

29 
28 

27 
26 

6 

26 

2-6 

26 

2.6 

25 

2-5 

35 
36 

^ 

39 

9-63  53d 
9-63  557 
9-63583 
9.  63  609 
9  63  636 

26 

26 
26 

26 

26 
-2 

9.68012 
9.68044 
9.68077 
9.68  109 
9.68  141 

32 
32 
32 
32 
32 

_;$ 

0.31  988 

0-31  955 
0.31  923 
0.31  891 
0.31  858 

9955^8 
9.95512 

9.95  5°6 
9-95  5o6 
9-95494 

6 
6 
6 
6 

25 
24 
23 

22 
21 

I 

9 

10 

20 

3-1 
3-5 

4.0 

4-4 
8-8 

3-0 
3-4 
3-9 
4-3 
8-6 

3-o 
3-4  : 
3-8 
4.2 

8-5 

40 

41 
42 

43 

44 

9.63  662 
9.63  683 
9.63715 
9.63  741 
9-6376? 

26 

26 

26 

26 

26 
*>f\ 

9.68  174 

9.68  20g 

9-68  238 
9.68  271 
9.68  303 

32 
32 
32 

H 

0.31  826 

0.31  793 
0.31  761 
0.31  729 
0.31  696 

9-95488 
9.95482 
9.95476 
9.95470 
9-95  464 

6 

6 
6 
6 

20 

19 
18 

17 
16 

3° 
40 

50 

I3-2 
17-6 

22.1 

13.0 
17.3 

21.6 

12.? 
17.0  i 
21.2 

45 
46 
47 
i  48 
49 

9-63793 
9-63819 
9.63  846 
9.63  872 
9.63  898 

26 

2§ 

26 
26 

iA 

9-68  335 
9.68  368 
9.68  400 
9.68432 
9.  68  464 

^ 
32 
32 
32 
32 

0.31  664 
0.31  632 
0.31  600 
0.31  56? 
0.31  535 

9.95458 
9-95452 
9-95  445 
9-95  439 
9-95433 

6 

6 
6 
6 

15 

14 
13 

12 
II 

6 

6 

0.6 

o.? 
~ 

6 

0.6 

0.7 

Q 

I 

0-5 

o.£ 

50 

5i 
52 
53 

54 

9.63924 
9.63956 

9-63976 
9.64002 

9.64023 

20 
26 

26 
26 
26 

_/- 

9.68497 
9.68  529 
9.68  561 
9.68  593 
9.68  625 

32 
32 

11 

32 

0.31  503 
0.31471 

0.31  439 
0.31406 
0.31  374 

9.9542? 

9-95  42i 
9.95415 

9-95  409 
9-95403 

6 

6 
6 
6 

10 

6 

9 

10 
20 

30 
AO 

o.s 

I.O 

I.I 

2.1 
3-2 

O.  O 

0.9 

I.O 
2.0 

3-0 

0-? 

0.8 
0.9 
1-8 

2"? 

[I 

57 
58 

59 

9.64054 
9.64086 
9.64  105 
9.64  132 
9.64153 

20 

26 
26 
26 

26 

9.6865? 
9.68  690 
9.68  722 
9.68  754 
9.68  786 

32 
32 
32 
32 
32 

0.31  342 
0.31  310 
0.31  278 
0.31  246 
0.31  214 

9-95397 
9-95390 
9-95  384 
9-95  378 
9-95  372 

6 
6 
6 
6 

5 
4 
3 

2 
I 

qu 
50 

•3 
5-4 

4.<~> 

5.0 

4.6 

GO 

9.64  184 

2i> 

9.68818 

3* 

0.31  182 

9-95  366 

0 

Log.  Cos. 

i  d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

p. 

p. 

TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

26° 


/ 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p. 

P. 

0 

! 

2 

3 
4 

9.64  184 
9.64  2IO 
9.64236 
9.64  262 
9.64287 

26 
26 
26 

25 
0/r 

9.68818 
9.68850 
9.68882 
9.68  914! 

9-68  946 

1 

32 
32 

0.31  182 
0.31  150 
0.31  II? 
0.31  085 
0.31053 

9.95366 
9.95  360 

9-95  353 
9-9534? 
9-95  34i 

6 

6 
6 
6 

2 

60 

59 
58 

i 

1 

8 
9 

9-64  3J3 
9-64339 
9.64365 
9.64391 
9.64416 

26 

2 

25 

_zr 

9-68978 
9.69016 
9.69042 
9.69074 
9.69  1  06 

32 
32 

ft 

32 

0.31  021 
O.3O  989 
0.3095? 
0.30926 
0.30894 

9-95335 
9-95  329 
9-95323 
9-95316 
9.95316 

6 
6 
6 

1 

* 

55 
54 
53 
52 
5i 

6 

8 
9 

32 

3-: 
3-i 

4-: 

4-< 

3 
z      ^ 
J     2 
5     4 
?     4 

2 
.2 

•? 

2 

:.s 

l    10 

ii 

12 
13 
14 

9.64442 
9.  64  468 

9-64493 
9.64519 

9.64545 

20 

25 
25 
26 

25 
i? 

9.69  138 
9.69  170 
9.69  202 
9.69234 
9.69265 

32 
32 
32 

8 

0.30  862 
0.30830 
0.30798 
0.30766 
0.30734 

9.95304 
9.95298 

9-95292 
9-95  285 
9-95  279 

i 

6 

2 

50 

49 
48 
47 
46 

10 

20 

30 
40 

50 

^ 

10.1 

16.: 

2I.< 

27.1 

\-     5 
5    ic 

5     It 

;  21 

2^ 

-3 

'-6 

.0 

:|    ; 

ii 

17 

18 

19 

9.64  570 
9.64  596 
9.64622 

9-6464? 
9.64673 

^ 

11 

25 

25 
/,? 

9.6929? 
9.69329 
9.69  361 

9-69393 
9.69425 

32 
32 

3* 

32 
32 

•3T 

o.  30  702 

0.30  676 
0.30  639 

o.  30  607 

0.30575 

9-95273 
9.95  267 
9.95  266 
9-95  254 
9-95  248 

6 
6 

g 

6 

6 

6" 

45 
44 
43 

42 

4i 

3i 

3 

i 

.  20 

21 

22 
23 
i      24 

9.64693 
9.64724 
9.64749 
9.64775 
9.64800 

25 
25 
25 
25 

25 

2p 

9.69455 
9.69483 
9.69  520 
9.69552 
9-69583 

31 
32 

3? 

i? 

0.30543 

o.  30  5  1  T 
o.  30  480 
o.  30  448 
0.30415 

9.95242 
9-95  235 
9-95  229 
9-95  223 
9-952I7 

1 

6 
6 
? 

40 

39 
38 
37 
36 

6 

8 
9 

10 

3-i 
3-7 
4.2 

4-7 

5.2 

3 
3 
4 
4 
5 

.1 

.6 
.1 

•6 

25 
26 

27 
28 
29 

9.64826 
9.64851 
9.64875 
9.64902 
9-6492? 

5 
25 
25 

25 
25 

9.69615 
9.69  647 

9-69678 
9.69716 
9.69742 

32 

il 

32 
31 

IT* 

0.30384 

0.30353 
0.30321 
0.30289 
0.30258 

9.95216 

9-95  204 
9.95  198 

9-95  191 
9-95  i8§ 

6 
6 

6 
6 
6 

35 
34 
33 
32 
3i 

20 
30 
40 
50 

lu.ij 

15.? 

21.  C 
26.2 

10 

15 

20 
25 

•3 

J 

-8 

30 

3i 

32 
33 
!    34 

9.64952 
9.64978 
9.65003 
9.65  02§ 

9.65054 

25 
25 
2f 
25 
25 

9.69773 
9.69  805 
9.69837 
9.69863 
9.69900 

31 
32 
3? 

$ 

_» 

0.30225 
0.30  194 
0.30  163 
0.30  131 
0.30  100 

9-95  179 
9-95  i?3 
9-95  165 
9.95  1  60 
9-95  154 

6 
6 

6 

30 

29 

27 
26 

6     : 

26 

5.6 

25 

2-5 

25 

2-5 

35 
36 
37 
38 
39 

9.65  079 

9.65  104 

9.65  129 
9.65155 

9.65  180 

25 

2§ 
25 
25 
25 

9.69931 
9.69963 
9.69994 
9.70025 
9.70058 

31 
3? 
3? 
32 
3? 

_c 

o.  30  o6§ 
0.30037 
0.30005 
0.29973 
o.  29  942 

9-95  H? 
9-95  HI 
9-95  135 
9.95  128 
9.95  122 

6 

6 
6 

6 
6 

25 
24 
23 

22 
21 

1  \ 

9    : 

10       i 
20       i 

J.O 

5-4 
J-9 

L-3 
5-6 

3-o 
3-4 
3-8 
4.2 
8.5 

2£ 

2.9 
3-3 

11 

8.3 

40 

4i 
42 
43 
44 

9.65  205 
9-65236 
9.65255 
9.65  286 
9.65  305 

25 
25 
25 
25 
25 
o? 

9.70089 

9.70  121 
9.70152 
9.70183 
9.70215 

31 
3? 
3i 

3 

0.29  916 
0.29  879 

0.2984? 
0.29815 
0.29785 

9.95  116 
9-95  109 
9-95  103 
9.95097 
9.95096 

6 
& 
6 

6 

20 

19 

18 

17 
16 

3°   i. 
40  i; 

50  21 

.0    C 

'-3   i 
•6  2 

•  7 
7-o 

1.2 

12.5 
16.5 

20.§ 

4I 
46 

47 
48 

49 

9-6533I 
9.65356 
9.65  381 
9.65  406 
9.65431 

25 
25 
25 

25 
25 

9.70246 
9.70278 
9.70309 
9.70341 
9.70372 

J1 
31 

3? 
3i 
3i 

0.29753 

0.29  722 
o.  29  696 
0.29  659 
0.29628 

9.95084 
9-95  078 
9.95071 
9.95065 
9-95058 

b 
6 

1 
g 
g 

15 
H 
13 

12 
II 

6 

24 
2.4 

2-8 

§ 

0.5 
0.7 

6 

0.6 
0.7 

50 

5i 

52 
53 
54 

9.65456 
9.65  481 
9.65  506 
9.65  536 
9-65555 

2S 
25 

3 

25 

9.70403 

9-70435 
9.70465 
9.7049? 
9.70529 

31 

11 

3i 
3i 

0.29595 
0.29565 

0.29  533 
0.29  502 
0.29471 

9.95052 
9-95046 
9-95039 
9-95  °33 
9.95025 

6 

6 

g 

10 

i 

6 

8 
9 

10 
20 
30 

3-2 

3.7 

4.1 

8.1 

12.2 
TA  5 

0.8 

I.O 

i.i 

2.T 

21 

0.8 
0.9 

I.O 
2.0 

3-0 

55 
56 

P 

59 

9.65  586 
9.65605 
9.65  630 
9.65655 
9.65  680 

2i> 
25 
24 
25 
25 

0  ? 

9.70566 
9.70  591 
9.70623 
9.70654 
9.70685 

JA 

3^ 

31 

3? 

0.29439 
0.29403 
0.29377 
o.  29  346 
0.29314 

9.95  020 
9.95014 

9.95  oo? 
9.95  ooi 
9-94994 

6 
6 
6 
6 

5 
4 
3 

2 

4<j 

50 

ID-3 
20.4 

4.3 
5-4 

4.0 
5.0 

60 

9-65  704 

24 

9.70715 

31 

0.29283 

9.94988 

6 

0 

Log.  Cos. 

<K 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

r 

P.  J 

>^ 

63° 


374 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

27° 


/ 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

P. 

p. 

0 

I 

2 

3 

4 

9.65  704 
9.65  72§ 
9-65754 

9.65  779 

9.65  803 

25 

24 

25 

24 

9.70715 
9.70748 
9.70779 
9.70810 
9.7084! 

31 
31 
31 
31 

0.29283 
0.29252 
0.29  221 
0.29  IQO 

0.29  158 

9.94988 
9.9498! 

9-94975 

9.94969 
9.94962 

6 
6 

2 

60 

P 

% 

i 

8 

i  -9 

9-65  823 
9.65  853 
9.65  878 
9.65  902 
9.65  927 

25 

24 

25 
24 
24 

X,? 

9.70872 
9.70903 

9-70935 
9.  70  966 
9.70997 

31 
3i 

3i 
3i 
3i 

0.29  12? 
0.29095 
0.29065 
0.29034 

o.  29  003 

9.94956 

9-94949 
994943 
9-94936 
9-94930 

O 

6 
6 

55 
54 
53 
52 
5i 

6 

8 
9 

3l 

3-1 

3-7 
4.2 
4-7 

3* 

11 

4.1 
4-6 

30 

3-6 

3-5 

4.6 

4-6 

10 

ii 

12 
13 
14 

9.65951 

9-65  976 
9.66001 
9.66025 
9.66050 

24 

a 

24 
24 

?T 

9.71  028 
9.71  059 
9.71  090 
9.71  12? 
9.71  152 

31 
3i 
3i 
3i 
3i 

0.28  972 
0.28  946 
0.28  909 

0.28878 
0.2884? 

9  94  923 
9.94917 
9.94916 
9-94904 
9-9489? 

\ 

50 

49 
48 
47 
46 

10 

20 

30 
40 

50 

!>-2 
10.5 

I5-? 

21.0 
26.2 

i>-A 
10.3 

15-5 
20.5 
25-8 

& 

i5.2 
20.3 

25.4 

i  ii 

17 

18 
19 

9.66074 
9.66099 
9.66  123 
9.66  148 
9.66  172 

•^4 
24 
24 
24 

24 

•7/f 

9.71  183 

9.71  214 

9.71  24$ 

9.71  275 

971  307 

31 
3i 
31 
31 
31 

0.28815 
0.28785 
0.28754 
0.28723 
0.28692 

9.94891 
9-94884 
9.94878 
9.94871 
9-94865 

6 
6' 

45 
44 
43 

42 
4i 

25 

20 

21 

22 

23 
i      24 

9.66  197 

9.66  221 
9.66  246 
9.66  276 
9.66294 

*4 

24 
24 
24 
24 

9-7i  338 
9.71  369 
9.71  400 

9.71  43i 
9.71  462 

31 
31 
31 
31 
3i 

0.28661 
0.28636 
0.28  599 
0.28  568 
0.2853? 

9.94858 
9-94852 
9.94845 
9-948^9 
9-94832 

6 
6 
6 
6 
6 

40 

39 
38 

% 

6 

8 
9 

10 

2-5 

2.9 

3.3 
3.? 

4.1 

85 

11 

27 
28 
29 

9.66319 

9-66  343 
9.66  36? 
9.66  392 
9.66415 

24 
2^ 
24 

24 

24 

9-71  493 
9.71  524 

9-7i  555 
9.71  586 
9.71  617 

31 
30 
31 
3i 
31 

0.28  rog 
0.28476 
0.28445 
0.28414 
0.28  383 

9-94825 
9.94819 
9.94812 
9.94806 
9-94799 

7 
6 
6 
6 
6 

35 

34 
33 
32 
3i 

20 

30 
40 

50 

•3 
12.5 

16.5 
20.  § 

80 

3i 
32 
33 
34 

9.66446 
9.66465 
9.66489 
9.66513 
9-66  537 

24 

24 

24 
24 
24 

9-7164? 
9.71  678 
9.71  709 
9.71  740 
9.71  771 

3U 
3i 
31 

3° 
31 
•20 

0.28  352 
0.28  321 
0.28  296 
o.  28  260 
0.28  229 

9-94793 
9-94786 
9-94779 
9-94773 
9-9476§ 

6 
6 

I 

6 
? 

30 

29 

28 

27 
26 

6 

24 

2.4 

24 

2.4 

23 

2-3 

35 
36 

% 

39 

9.66  561 
9.66  586 
9.66610 
Q.  66  634 
9  66653 

24 

24 
24 

24 
24 

9.71  801 
9-71  832 
9.71  863 
9.71  894 

9.71  925 

3° 
3i 
3i 
30 
3i 

•70 

0.28  198 
0.28  1  6? 
0.28  135 
0.28  106 

0.28  075 

9.94760 
9-94753 
994746 
9.94740 

9-94733 

b 
6 

I 
6 

2 

25 
24 
23 

22 
21 

8 

9 

10 

20 

2.8 

3-2 
3-7 
4.1 
8.T 

2.8 

3.2 

3.6 

4.0 

8.0 

2.? 

3-t 
3-5 
3-? 

7-8 

T    T       fj 

10 

41 
42 
43 
i    44 

9.66682 
9.66  705 

9.66730 
9.66754 

9-66778 

24 

24 
24 
24 

24 

9-7i  953 
9.71  986 
9.72017 
9-7204? 
9-72078 

3° 

30 

% 

31 

o.  28  044 
0.28014 
0.27983 
0.27  952 
0.27  921 

9.94727 
9.94726 
9-947I3 
9.94707 
9.94706 

6 
6 
7 

& 

6 

20 

J  Q 

I  O 

17 

16 

3° 
40 

50 

16.3 
20.4 

16.0 

20.0 

ii.  7 

15-6 
19.6 

45 
46 

47 
!    48 
49 

9.66802 
9.66825 
9.66  856 
9.66874 
9-66893 

24 

24 
24 

24 

24 

9.72  109 
9.72  139 
9.72  170 

9.72  201 
972231 

3° 
33 
33 
3i 
30 

-•>A 

0.27  891 
0.27866 
0.27  830 
0.27799 
0.27768 

9-94693 
9-94687 
9.  94  686 
9.94674 
9.94667 

'i 
6 
6 
6 
7 

15 
14 
13 

12 
II 

6 

7 

0.7 
0.8 

& 

0.5 

0.? 

6 

0.6 
0.7 

0 

50 

5i 
52 
53 
54 

9.66  922 
9.66945 
9.66  976 
9.66994 
9  67018 

24 

24 
24 
23 
24 

9.72  262 
9.72  292 
9-72323 
9-72354 
9.72  384 

3° 

30 
30 

? 

0.27738 
0.2770? 
0.27  677 
0.27  646 
0.27615 

9.94666 
9.94654 
9.94647 
9.94646 
9-94633 

6 
6 
7 
6 

7 

10 

7 
6 

9 

10 

20 

1        30 
A.O 

0.9 
1.6 
i.i 

2-3 
3-5 

A  Z 

0.8 

I.O 

I.I 

2.  1 

ll 

o.o 

0.9 

I.O 
2.0 

3-o 

40 

II 
P 

59 

9.67  042 
9.67066 
9.67089 
9.67  113 
9.67  137 

24 

24 

23 

24 

23 

9.72415 
9.72445 
9.72476 
9.72  505 
9-72537 

3° 
33 
35 
30 
30 

0.27585 
0.27  554 
0.27  524 
0.27493 
0.27463 

9.94627 
9.94  626 
9.94613 
9-94607 
9.94600 

6 
"6 
7 
6 
7 

5 
4 
3 

2 

I 

50 

^-•o 
5-8 

4-  J 
5-4 

.<-» 

5.0 

60 

9.67  161 

24 

9.72  56? 

3° 

0.27432 

9-94  593 

6 

0 

Log.  Cos. 

(1. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

p 

P. 

375 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

28° 


Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p 

8 

,  P. 

0 
I 

2 

3 

!      4 

9.67  161 
9.67  184! 
9.67203 
9.67  232 
9.67256 

23 
24 
23 
24 

9.7256? 
9.72  598 
9.72623 
9.72659 
9.72689 

30 

3? 
30 

30 

or» 

0.27432 
0.27402 
0.2737! 
0.27341 
0.273II 

9-94  593 
9.94  587 

9-94  580 
9-94  573 
9-94  566 

6 

7 
6 
7 

? 

60 

59 
58 

6 

8 
9 

9.67  279 
9.67  303 
9.67327 
9-67  350 
9-67  374 

23 
23 
24 
23 
23 

9.72719 
9.72750 
9.72786 

9.72  811 
9.72841 

3° 
30 

33 
30 

0.27  286 
0.27  250 
0.27  2I§ 
0.27  189 
0.27  159 

9.94  560 
994553 
9-94546 
9-94539 
9-94533 

6 

6 
6 

55 
54 
53 
52 

30 

6     3.6 

7     3-5 
8     4.6 

9    4-6 

30 

3-5 
4.0 

4-5 

29 

2-9 

3-4 
3-9 
4-4 

10 

ii 

1  2 
13 

9.67  39? 
9.6742! 

9.67445 
9.67463 

9-67492 

24 
23 
23 
23 

27i 

9.7287! 
9.72902 
9.72932 
9.72962 
9.72993 

30 
30 

30 
30 
30 

0.27  123 
0.27098 
0.2706? 
0.2703? 
0.27007 

9-94  526 
9.94519 
9.94512 
9.94506 
9-94499 

6 

7 

6 

7 

50 

49 
48 
47 
46 

10     5.1 
20  10.  ! 
30  15.2 
40  20.3 
50  25.4 

5.0 

10.0 

15.0 

20.0 
25.0 

4-9 

9-8 
14.7 

19-6 
24.6 

i  ii 

17 

18 

i   19 

9.67515 

9-67  539 
9.67  562 
9.67  586 
9.  67  609 

3 

23 
23 
23 
23 

o5 

9.73023 
9.73053 
9-73084 
9-73  IJ4 
9-73  I4-? 

30 
30 
30 
30 

0.26976 
0.26946 
0.26916 
0.26886 
0.26855 

9.94492 
9.94485 

9-94478 
9.94472 
9.94465 

6 
7 
6 
7 

45 
44 
43 
42 

24 

j    20 

21 

22 

23 

i      24 

967633 
9.67  656 
9.67679 
9-67703 
9.67  726 

23 

23 
23 

23 
23 
oo 

9-73  *74 
9-73205 

9-73235 
9.73265 

3° 
30 
3°. 

30 

0.26825 
0.26795 
0.26765 
0.26734 
o.  26  704 

9.94458 
9.9445! 
9.94444 
9-9443? 
9-94431 

6 
7 

I 

40 

39 

38 

6 

7 

8 

9 

10 

2.4 

2.8 

4.0 

26 

27 
28 

29 

9.67750 
9.67773 
9-67796 
9.67819 

9,67843 

23 
23 

23 

23 

9-73  325 
9-73356 
9.73386 
9.73416 
9-73446 

3° 
33 
30 

3° 
30 

0.26674 
o.  26  644 
0.26  614 
0.26  584 

0.26553 

9-94424 
9.94417 
9.94416 
9.94403 
9-94396 

I 
7 
7 
2 

35 
34 
33 

20 
30 
40 

5° 

.0 
12.0 

16.0 

20.0 

30 

32 
33 
1    34 

9.67  866 
9.67889 
9.67913 
9.67936 
9.67959 

23 

23 
23 
23 

9-73476 
9-73506 
9-73  536 
9-73  567 
9-73  597 

3° 
30 
30 
33 
3° 

0.26523 
0.26493 

o.  26  463 

0.26433 

o.  26  403 

9-94390 
9.94383 
9-94  376 
9.94369 
9.94  362 

6 
7 
7 
6 
7 

30 

29 
28 

27 
26 

23 

6     2.3 

23 

2.3 

22 

2.2 

35 
i    36 

i    39 

9.67  982 
9.68005 
9.68029 
9.68052 
9.68075 

23 
23 
23 
23 

9.73627 

9.73657 
9.73687 

9-737I7 

9-73747 

3° 
30 
30 
30 
30 

0.26373 
0.26343 
0.26313 
0.26283 
0.26  253 

9-94355 
9-94348 
9-94341 
9-94335 
9.94328 

7 

6 

7 

25 
24 
23 

22 
21 

7     2.? 
8     3.1 
9     3-5 
10     3.9 
20     7.3 

2.7 

3-4 
3-| 

7-6 

2.6    | 

3-° 

31   ! 

3-? 

7-5   | 

40 

42 
43 
44 

9.68098 
9.68  12! 
9.68  144 
9.68  1  6? 
9.68  196 

23 
23 
23 
23 

9-73777 
9.73807 

9-73837 
9-73867 
9-73897 

3° 
30 
30 
30 
30 

0.26  223 
0.26  193 
0.26  163 
0.26  133 
0.26  103 

9.94321 
9-943J4 
9-94307 
9.94306 

7 
7 
6' 

7 

20 

19 

18 

17 
16 

4o  15-6 
50   19.6 

15-5 
19.! 

15.0 

18.?   | 

45 
46 
47 
48 
49 

9.68213 
9-68236 
9  68  259 
9.68  282 
9  68  305 

23 
23 
23 
23 

9.73927 
9.73957 

9.740I7 
9.74047 

30 
30 
30 
30 

0.26073 
o.  26  043 
0.26013 

0.25983 
0.25953 

9.94286 
9-94279 

9.94265 
9-94258 

7 
7 
7 
7 

15 
14 
13 

12 
II 

6  c 

7   c 

7      « 

•7    c 
.8   c 

3 

•6 

.? 

50 

52 
53 

54 

9.68323 
9.68351 
9-68374 
9-68397 
9.68  420 

23 

11 

23 

9-74076 
9.74  106 

9-74I36 
9-74  166 
9-74  196 

29 
30 
30 
30 
29 

0.25923 
0.25893 
0.25  863 
0.25833 

0.25  804 

9.9425! 

9-94  245 
9.94238 
9.94231 
9.94224 

7 
6 
7 

7 
7 

10 

9 
8 

7 
6 

c 
9   i 

10     I 

20     2 

30   3 

AO    A 

.9   c 

.!    i 
•3    2 

•5  3 

2      A 

•8 

.0 

.1 
.T 

.2 
5 

55 
56 
57 
58 
59 

9.68443 
9.68466 
9.6848^ 
9.6851! 
9-68  534 

23 

22 

23 
23 

9.74226 
9.74256 
9.74286 
9-743I5 
9-74345 

30 
30 
29 
30 

0.25774 
0.25744 
0.25714 

0.25  684: 

0.25654 

9.94217 

9.94  210 
9.94203 

9-94  196 
9-94  189 

7 
7 
7 
7 

5 
4 
3 

2 
I 

5°  5 

•O     *t 

•8    5 

•  j 

•4 

|    60 

9.68  557 

00 

22 

9-74375 

29 

0.25  625 

9.94  182 

7 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

' 

p 

.  P. 

61 


376 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

29° 


r 

LOST.  Sin. 

d, 

LOST.  Tan. 

o.  d. 

LO*.  Cot. 

Log.  Co«. 

d. 

p. 

p. 

0 
I 

2 

3 
4 

9.68  557 
9.68  580 
9.68602 
9.68625 
9.68648 

11 

23 

22 

9-74375 
9.74405 

9-74435 
9.74464 

9-74494 

30 
30 

29 

30 

0.25  625 
0.25595 
0,25  565 

0-25535 
0.25  505 

9.94  182 
9.94175 

9.94  1  68 
9.94  161 
9.94154 

7 
7 
7 
7 

00 

59 
58 

I 

6 

8 
9 

9.68671 
9-68693 
9-68  715 
9.68  739 
9/58  761 

23 

x>'> 

23 

22 
22 

9-74  524 
9-74554 
9-74  583 
9.74613 

9-74643 

^9 

3? 
29 
29 
30 

ori 

0.25476 
0.25  446 
0.25415 
0.25387 
0.25357 

9  94  147 
9-94  HO 
9.94I33 
9.94126 

9.94118 

7 
7 
7 

55 
54 
53 
52 
5i 

30 

6     3.0 

7     3-5 
8     4.0 

9     4-5 

29 

2-9 

3-4 
3-9 
44 

29 

2.9 

3-4 

3-8   i 
4-3  : 

10 

ii 

12 

13 
U 

9.68  784 
9.68807 
9.68829 
9.68  852 
9.68  874 

23 

22 
22 
22 
22 

9.74672 
9.74702 

9-74732 
9.74761 

9.74791 

29 
30 
29 
2§ 

30 
on 

0.2532? 
0.25  29? 
0.25  268 
0.25  238 
0.25  208 

9.9411! 

9-94  104 
9.9409? 
9.94096 
9.94083 

7 
7 
7 
7 

50 

49 
48 
47 
46 

10     5.0 

20    IO.O 
30    15.0 

40  20.  o 

50  25.0 

4.9 

9-8 
14.? 

19.6 
24.6 

4-8 
9-6 

14.5  ; 

19-3   i 
24.! 

II 

17 

18 
19 

9.68897 
9.68  920 
9.68  942 
9.68  965 
9.68  987 

23 

22 
22 
22 

00 

9.74821 
9.74856 
9.74880 
9.74909 
9-74939 

29 
29 
29 
29 
30 

0.25  179 
0.25  149 

0.25  120 
0.25  096 
0.25  066 

9-94076 
9.94069 
9.94062 
9.94055 
9.94048 

7 

7 
7 

45 
44 
43 
42 

4i 

23 

20 

21 
22 

23 
24 

9.69010 
9.69032 
9.69055 
9.69077 
9.69099 

22 
22 
22 
22 
~3 

9.74969 

9-74998 
9.75028 

9.7505? 
9.75087 

29 

29 
29 
29 
29 
on 

0.25  031 

0.25  oo! 
0.24972 
0.24942 
0.24913 

9.94041 
9.94034 
9.94025 
9.94019 
9.94012 

7 

7 
7 

40 

38 
37 
36 

6 

8 
9 

10 

2.3 
2.7 

3-5 
3-4 

3-8 

72 

25 
26 

27 
28 

29 

9.69  122 
9.69  144 
9.69  167 
9.69  189 
9.69211 

22 
22 
22 
22 

00 

9-75  H6 
9-75  H6 
9-75  175 
9-75  205 
9-75  234 

29 
29 
29 
29 
29 

0.24883 
0.24854 
0.24824 
0.24795 
0.24765 

9.94005 
9.93998 
9.93991 
9.93984 
9-93977 

9 

7 
7 
7 

a 

35 
34 
33 
32 
3' 

30 
40 

50 

•6 
ii.  5 
15-3 
19.1 

30 

3i 
32 
33 
34 

9.69  234 
9.69  256 
9.69273 
9.69  301 
9.69323 

22 
22 
22 
22 

9.75  264 
9-75293 
9-75323 
9-75352 
9.75  382 

2§ 
2§ 
29 
29 
29 

0.24736 
0.24705 
0.24677 
o.  24  64? 
o.  24  6  1  8 

9.93969 
9-93962 

9-93955 
9.93948 

9-93941 

7 

7 
7 

7 

30 

29 

28 

27 
26 

22 

6       2.2 

22 

2.2 

21 
2.T 

P 

i 

39 

969345 
9.69  367 
9.69390 
9.69412 
9.69434 

22 
22 
22 
22 
22 

9.75411 
9.75441 
9.75476 

9-75499 
9.75  529 

29 
29 
29 
29 
2§ 

?o 

0.24583 
0.24559 
0.24529 
o.  24  506 
0.24471 

9-93934 
9.93926 
9-939I9 
9.93912 

9-93905 

1 
7 
7 

25 
24 
23 

22 
21 

7     2.6 
8     3.0 
9     3-4 
10     3.7 

20       7.5 
7O      T  T    5 

2.5 
2.§ 
3-3 

3-8 

7.3 

no 

2.5 

2-8 

ll 

10  9 

40 

4i 

42 

43 
44 

9-69456 

9-69478 
9.69  506 

9-69523 
9.69  545 

22 
22 
22 
22 

22 

9-75  558 
9-75  588 
9.75617 

9.75646 
9.75676 

29 

2§ 

29 
29 
2§ 

0.24441 
0.24412 
0.24383 
0.24353 
0.24  324 

9.93898 
9.93891 
9-93883 

9-93  876 
9.93  869 

7 
7 

20 

'9 
18 

i? 
16 

'40      15.0 
50      I8.f 

14-6 
18.3 

L\J.  / 

14.3 
17.9 

41 
46 

47 
48 

49 

9.69567 

9-69  589 
9.69611 

9-69633 
9.69655 

22 
22 
22 
22 
22 

9.75705 

9-75734 
9.75764 

9-75  793 
9.75  822 

29 
29 
29 
29 

29 

0.24295 
0.24265 
0.24  236 
o.  24  207 

0.2417? 

9.93  862 
9.93854 
9-93  84? 
9.93840 

9-93833 

1 
7 

7 

i5 
14 

13 

I2> 

II 

6   c 
7   c 

1     7 

>.?  o. 
>-9  o. 

8 

50 

5i 
52 
53 

54 

9.69  67? 
9.69699 
9.6972! 
9.69743 
9.69765 

22 
22 
22 
22 
22 

9.75851 

9.75881 
9.75910 

9-75  939 
9.75968 

29 

29 

2? 
29 
29 

0.24  I4§ 
0.24  119 
o.  24  090 
0.24066 
0.2403! 

9.93  826 

9.93818 
9.9381! 
9.93  804 
9-93796 

7 

f 
7 

1 

10 

9 
8 

6 

i 
9    i 

10     1 

20    2 

30    2 
AQ      C 

.0  o. 
.1    i. 

.2     I. 

•5  2. 
,•?  3- 

O    A. 

I 
3 

t 

P 
P 

59 

9.69  78? 
9.69809 
9.69  831 
9.69  853 
9.69875 

22 
22 
22 
21 
22 

9.75998 
9.76027 
9.76056 
9.76085 
9.76  115 

29 
29 

29 
29 
2§ 

0.  24  002 
0.23973 

0.23943 
0.23914 
0.23885 

9-93789 
9.93782 

9-93775 
9.93767 
9.93766 

7 

1 
7 

7 

e 

5 
4 
3 

2 
I 

50  t 

>.§  5. 

8 

GO 

9.69897 

22 

9.76  144 

29 

0.23  856 

9-93753 

; 

0 

Log.  Cos. 

d. 

Log.  Cot. 

>c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

P 

p. 

60 


377 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

3O° 


f 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p. 

p. 

0 

I 

2 

3 

4 

9.69897 
9.69919 
9.69  946 
9.69962 
9.69984 

22 
21 

22 
22 

9.76  144 

9.76  173 

9.  76  202 
9.76231 
9.76  266 

29 
29 
29 
29 

0.23  856 
0.23  827 

0.2379? 
0.23768 
0.23739 

9-93753 
9-93746 
9-93738 
9-93  73i 
9.93724 

7 

7 
a 

00 

59 
58 

I 

I 

8 
9 

9.  70  006 
9.70028 
9.70050 
9.7007! 
9.70093 

22 
22 
21 

22 

9.76289 
9.76319 
9.76348 
9.76377 
9.  76  406 

29 

29 
29 
29 

0.23716 
0.23681 

0.23652 
0.23  623 

0.23594 

9-93716 
9.93709 
9.93702 
9.93694 
9.93687 

7 
7 

55 
54 
53 
52 

60   o 

29 

28 

o  $ 

10 

1  1 

12 

13 

14 

9.70115 
9.70137 
9.70153 
9.70  1  80 
9.  70  2O2 

22 
2! 
21 
22 

9-76435 
9.76464 
9.76493 
9.76  522 
9.7655! 

29 
29 
29 
29 
29 

0.23565 
0.23535 
0.23505 
0.2347? 

0.23448 

9.93680 
9.93672 
9.93665 
9.93658 
9.93656 

7 

7 

50 

49 
48 

47 
46 

2.9 

7     3-4 
8     3-9 
9     4-4 
10     4.9 

20      9.8 

•9 
3-4 
3-8 
4-3 
4-8 
9-6 

2-8     1 

3-8 
4-3 
4-? 
9-5 

17 

18 
19 

9.70223 
9.70245 
9.70267 
9.70288 
9.70310 

21 
22 
21 
21 

9.76586 
9.76609 
9.76638 
9.7666? 
9.76695 

29 
29 
29 
29 

29 

0.23419 
0.23396 
0.23  36! 
0.23332 
0.23303 

9-93643 
9-93635 
9-93628 
9-93621 

7 

? 

45 
44 
43 
42 

30  14.? 
40  19-6 
50  24.6 

14-5 
19.3 
24.1 

14.2 
19.0 

23-? 

20 

21 

22 
23 
24 

9-7033I 
9.70353 
9-70375 
9-70396 
9.70418 

22 
21 
21 
21 
of 

9.76725 
976754 
9.76783 
9.76812 
9.76  84! 

29 
29 
29 
29 
29 

0.23  274 
0.23245 
0.23215 
0.23  1  8? 
0.23  158 

9.93606 
9-93599 
9-93591 
9-93584 
9-93  576 

7 

? 
R 

40 

39 
38 

i 

25 
26 
27 
28 
29 

9-70439 
9.70461 
9.70482 
9.70504 
9.70525 

21 
21 
21 
21 
_o 

9.76  876 
9.76899 
9.76928 
9.76957 
9.76986 

29 
28 

29 
29 
29 

0.23  129 
0.23  ioi 
0.23072 
0.23043 
0.23014 

9-93  569 
9-93  562 
9-93  554 
9-93  547 
9-93  539 

7 
7 

? 

^ 

35 
34 
33 
32 

22 

6      2.2 

7     2.5 
8     2.9 

21 

2.1 
2-5 
2-8 

21 

2.1 

24 

2.8 

30 

32 
33 
34 

9.70547 

9-70568 
9.70  590 
9.70611 
9.70632 

21 
21 
21 
21 
21 
_~ 

9.77015 
9-77043 
9.77072 

9.77  10! 
9.77  130 

29 

28 

29 
29 
29 

O.22  985 
0.22956 
0.2292? 
O.22  898 
0.2286§ 

9-93  532 
9-93524 
9.93517 
9-93  5o§ 
9-93  502 

7 

? 

? 

30 

29 
28 

27 
26 

9     3-3 
10     3-6 
20     7.3 
30  n.o 
40   14.5 
co  18  3 

H 

10.? 

14-3 
I  7  Q 

3-1 
3-5 
7.0 
10.5 
14.0 

17   £ 

$ 

37 
38 
39 

9.70654 
9.70675 
9.70696 
9.70718 
9.70739 

21 
21 
21 
21 

21 

9.77  159 
9.77  1  88 
9.77217 
9-77245 
9-77274 

29 
29 

28 

29 

0.22  841 
0.22812 
0.22  783 

0.22754 
0.22725 

9-93495 
9-9348? 
9.93480 
9-93472 
9.93465 

7 

j 

« 

25 
24 

23 

22 
21 

i/.y 

17*5 

40 

42 
43 
44 

9.70766 
9.70782 
9.70803 
9.70824 
9.70846 

21 

21 
21 
21 

21 

9-77  303 
9-77  332 
9.77  361 

9-77  389 
9-77418 

28 

29 

28 

29 

0.22  695 
0.22668 
0.22  639 
0.22  6l6 
0.22  58! 

9-9345? 
9-93450 
9-93442 

9-93435 
9.9342? 

7 

? 

0 

20 

19 
18 

17 
16 

8 

6   0.8 

i 

7 
0.7 

45 
46 
47 
48 

49 

9.70  867 
9.70888 
9.70909 
9.70936 
9.70952 

21 
21 
21 
21 
2! 

9-77447 
9.77476 
9-77  504 
9-77  533 
9  77  562 

29 

28 

29 

0.22  553 

0.22  524 
0.22495 
0.22465 
0.22438 

9.93420 
9.93412 
9.93405 
9.9339? 
9-93390 

7 
? 

15 
13 

12 
II 

7   0.9 
8    i.o 

9     1.2 

10    1.3 

20     2.6 

0.9 

I.O 

I.I 

1.2 
2.5 

0.8 
o«9 

£j 

50 

52 
53 

54 

9.70973 
9.70994 
9.7IOI5 

9-71  036 
9.71  05? 

21 

21 
2l 
21 
21 

9-77  59i 
9-77619 
9.77648 
9.77677 
9-77  7o5 

2§ 
28 

29 

28 

0.22409 
0.22  386 
0.22352 
0.22323 
0.22  294 

9-93  382 
9-93  374 
9-93  367 
9.93  359 
9-93352 

? 

? 
P 

10 

6 

30    4.0 

40    5-3 
50   6.g 

3-7 

5-o 

6.2 

3-5 
4-S 
5-8 

59 

9.71  078 
9.71  099 
9.71  121 
9.71  142 
9.71  163 

21 
21 
21 
21 
21 

9-77734 
9.77763 

9-77  79i 
9.77  820 
9.77849 

29 

28 

28 

29 

0.22  266 
0.22  237 
O.22  2O§ 
0.22  1  80 
0.22  151 

9-93  344 
9-93337 
9-93329 
9-93  32i 
9-933H 

7 

8 

p 

5 
4 
3 

2. 

I 

00 

9.71  184 

21 

9.7787? 

28 

0.22  122 

9-93306 

1 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

V 

.  P. 

59 


378 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

31° 


r 

I."--.  Siu. 

d. 

Loir.  Tan. 

r.  d. 

LOST.  Cot. 

Loe.  Cos. 

d. 

P 

P. 

? 

2 

3 
4 

9.71  184 
9.71  205 
9.71  226 
9.71  247 
9.71  268 

21 
21 
21 
21 

9.77877 
9.77906 

9-77934 
9-77  963 
9.77992 

28 

28 

28 

29 

0.22  122 

0.22094 
0^.22065 
0.22037 
0.22008 

9-93306 
9-93  299 
9-93  291 
9-93  284 
9.93276 

7 

9 

8 

ft 

00 

P 

P 

I 
I 

9 

9.71  289 
9.71  310 
9.71  331 
9-71  351 
9.71  372 

21 
21 
.20 
21 

9.78026 
9.78049 
9.78077 
9.78  106 
9-78I34 

25 

28 

28 
28 
28 

0.21  979 

0.21  951 
0.21  922 
0.21  894 

0.21  865 

9-93  263 
9.93261 

9-93  253 
9-93245 
9-93238 

/ 

I 

f 

a 

55 
54 
53 
52 
5i 

k29 

f.     ~  . 

2 

8 

- 

28 

-7    X 

10 

!J 

\l 

9-71  393 
9.71414 

9-7i  435 
9.71  456 

9-7i  477 

21 
20 
21 
21 

9.78  163 
9-78  191 

9.78  220 

9.78  248 

9.78277 

28 

28 
28 
28 

•70 

0.21  837 

0.21  808 

0.21  780 
0.21  751 
0.21  723 

9-93  236 
9.93223 
9.93215 
9-9320? 
9.93200 

7 

8 
9 

0 

50 

49 
48 

47 
46 

7     3- 
8     3-, 
9    4-. 
10    4., 

2O      Q.i 

J 
1 

1 
I 

3 
5 
4 

4 

f': 

8 
8 

J 

c 

it 

4.2 

4-6 
9-3 

15 
16 

17 
18 

19 

9.71498 
9-71  5^8 
9-7i  539 
9.71  560 
9.71  581 

2O 
21 
20 
21 

^0 

9.78305 

9.78  334 
9.78362 
9.78391 
9.78419 

28 

28 
28 
2§ 
28 

10 

0.21  694 

0.21  666 

0.21  637 
0.21  609 
0.21  586 

9-93  192 
9-93184 

9-93  177 
9-93  169 
9-93  1  61 

9 

8 

9 

Q 

45 
44 

43 
42 
4i 

30    14- 
40    19. 
50    24. 

5 

14 
19 

-3 

0 

7 

14.0 

i8.g 

23.3 

20 

21 

22 

Li 

9.71  6oT 
9.71  622 
9.71  643 
9.71  664 
9.71  684 

21 
20 
21 
20 

9.78448 
9-78476 
9-78505 
9.78533 
9.78  561 

28 

28 

28 

28 

28 
to 

0.21  552 
0.21  523 

0.21  495 

0.21  467 

0.21438 

9-93I53 
9-93  H6 
9-93  138 
9-93136 
9-93  123 

1 

8 

9 

40 

39 
38 

1 

!    25 
1    26 
27 
28 
29 

9.71  705 
9.71  726 
9-71  746 
9.71  767 
9.71  788 

20 
2O 
21 
20 
ir\ 

9.78590 
9-78613 
9.78647 
9.78675 
9.78703 

28 

28 
28 

28 

28 

O.2I  4IO 
0.21  38! 

0.21  353 

0.21  325 

0.21  295 

9-93II5 
9-93  10? 
9-93  ioo 
9.93092 
9.93084 

1 

8 

9 

35 
34 
33 
32 
3' 

21 

6       2. 

7     2. 

8       2. 

i 

$ 

s 

2 

2 
% 

2 

6 

.c 
•4 

20 

2.0 
2-3 

2-6 

30 

3i 

32 

[| 

9.71  8o§ 
9.71  829 
9.71  849 
9.71  870 
9.71  891 

20 
20 
21 
20 
oA 

9.78732 
9.78766 
9.78783 
9.78817 
9.78845 

-70 

28 

28 

28 

28 

28 
_Q 

0.21  268 
0.21  239 
0.21  211 
0.21   183 
0.21   154 

9-93076 
9.93069 
9.93061 
9.93053 
9-93045 

8 

8 

^ 

30 

29 
28 

27 
26 

9     3- 
10     3. 

20       7. 

30   10. 

40     14. 

5 

D 

5 

o 

3 

I 

1C 

13 

y  - 

•4 
-8 
.2 

-6 

3-0           ; 

10.0 

ll'l 

11 
IE 

39 

9.71  911 
9.71  932 
9.71  952 
9-7i  973 
9-7I993 

2O 
20 
20 
20 
20 

9-78873 
9.78902 
9.78930 
9-78958 
9-78987 

2o 

28 

28 

28 
28 

0.21   126 
0.21  098 
0.21  070 
0.21  04! 

0.21  013 

9-93038 
9.93030 
9-93022 
9.93014 
9-93006 

7 
8 

9 

8 
8 

25 
24 
23 

22 
21 

5°   17' 

*/ 

40 

4i 
42 
43 

44 

9.72014 
9.72034 
9.72055 
9.72075 
9.72096 

2O 
20 
20 
20 
20 

9.79015 

9-79043 
9.79071 
9.79  loo 

9.79128 

2o 

28 
28 

28 

28 

0.20985 

0.20956 
o.  20  923 
0.20900 
0.20872 

9.92999 
9.92  991 
9.92983 
9.92975 
9.9296? 

7 
8 

8 

8 

20 

18 

17 
16 

6 

c 

8 

.S 

f 

o. 

9 

45 
46 
47 
48 
49 

9.72  116 
9-72  136 
9-72I57 
9.72  177 
9.72  198 

20 
20 
20 
20 

9.79156 
9.79  184 
9.79213 
9.79241 
9.79269 

^ 
28 

28 

28 
28 

0.  20  843 
0.  20  8  1  5 
0.20787 
0.20759 
0.20731 

9.92  960 
9.92952 
9.92944 
9-92  936 
9-92928 

9 

8 
8 

8 

15 
14 
13 

12 
II 

I 

9 

10 
20 

c 

1 

1 

-9 
.6 

.2 

•3 

•6 

0. 

I. 
I. 

2. 

9 

0 

I 

2 

5 

50 

5i 
52 
53 

54 

9.72  218 
9.72  238 
9.72259 
9-72  279 
9.72299 

20 
20 
20 
20 

9.79297 
9-79325 

9-79354 
9.79382 
9.79410 

^8 

28 

28 

28 
28 

0.  20  702 
0.  20  674 
0.  20  646 
0.20618 
0.20  590 

9.92926 
9.92913 
9.92905 
9.92  897 
9.92  889 

I 

8 

9 

10 

6 

30 
40 
50 

4 

I 

.0 
-3 

•6 

3- 

t 

? 

0 

2 

H 
11 

59 

9.72319 
9.72340 
9.72  360 
9.72  386 
9.72406 

2O 
20 
20 
2O 

20 

9-79438 
9.79465 
9-79494 
9.79522 

9.79551 

^8 

28 
28 
28 

28 
_o 

0.20  56! 
0.20533 
0.20  505 

0.2047? 
o.  20  449 

9.92  88T 
9.92873 
9.92  865 
9.92858 
9.92  850 

8 
8 

8 

5 
4 
3 

2 
I 

60 

9.72421 

2O 

9-79579 

25 

0.20421 

9.92  842 

0 

Log.  Cos. 

d. 

Log.  Cot. 

0.  (1. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

P 

p. 

58 


379 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

32° 


t 

Log.  Sin. 

(1. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p. 

p. 

0 
I 

2 

3 
4 

9.72421 
9.72441 
9.7246! 
9.72481 
9.72  501 

20 
20 
20 
20 

9-79  579 
9.79607 

9-79635 
9.79663 
9.79691 

28 
28 
28 

28 

^0 

o.  20  42  1 

0.20393 

0.20365 

0.20337 

0.  20  30g 

9.92  842 
9.92834 
9.92826 
9.92813 

9.92  816 

8 

8 
8 

00 

ii 

i 

I 

8 
9 

9.72  522 
9-72542 
9.72  562 
9.72  582 
9.72602 

20 
20 
20 

20 

9.79719 
9-7974? 
9-79775 
9.79803 
9.79831 

28 
28 
28 
28 
~Q 

0.  20  286 
0.20  252 
0.20224 

0.20  195 

0.20  l6g 

9.92  802 
9.92794 
9.92786 
9.92  778 
9.92771 

8 
8 
8 

? 

55 
54 
53 
52 
5i 

28 

28 

2? 

10 

ii 

12 
13 
14 

9.72622 
9.72642 
9.72662 
9.72682 
9.72702 

20 
20 
20 
20 

9.79859 
9.79887 
9.79915 

9-79943 
9.79971 

28 
28 
28 
28 
?Q 

0.20  140 
O.2O  112 
0.  20  084 

0.20056 

0.2002§ 

9.92763 
9.92755 
9.92747 
9.92739 
9.92731 

8 
8 
8 
8 

50 

49 
48 

47 
46 

8 
9 

10 

20 

2-8 

11 

4-3 

4-? 

9c 

2.5 

l\ 

4.2 

4-6 

O   1 

2-? 

3-2 
3-6   ! 
4.1 
4-6 

0    T 

II 

17 

18 
19 

9.72723 

9-72  743 
9.72  763 
9.72783 
9.72802 

20 
20 
20 
19 

9.79999 
9.8002? 
9.80055 
9.  80  083 
9.  80  1  1  1 

28 
28 
28 
28 

?R 

0.20000 

0.19972 
0.19944 
0.19916 

o.i  9  88§ 

9.92723 
9.92715 
9.92707 
9/92  699 
9.92  691 

8 
8 
8 
8 
8 

45 
44 
43 
42 
4i 

30 
40 
50 

o 
14.2 
19.0 
23-? 

y.  j 

14.0 

iS.g 
23-3 

y.  i 

I3-? 
I8.§ 
22.9 

20 

21 

22 

23 

24 

9.72822 
9.72842 
9.72  862 
9.72882 
9.72  902 

20 
2O 
2O 
20 

9.80  139 
9.8016? 
9.80195 
9.80223 
9.80251 

28 
28 
28 
28 
«e 

o.  1  9  866 
0.19  832 
o.  1  9  804 
0.19776 
0.19748 

9.92  683 
9.92675 
9.92  667 
9.92659 
9.92651 

8 
8 
8 
8 

40 

39 
38 
37 
36 

3 

27 
28 
29 

9.72  922 
9.72942 
9.72  962 
9.72982 
9.73002 

19 
2O 
20 
20 

9.80279 
9.80307 

9-80335 
9.  80  363 
9.80391 

27 
28 
28 
28 
28 

x,ft 

0.19  721 
0.19693 
0.19  665 
0.19637 
0.19609 

9  92  643 
9.92635 
9.92627 
9.92619 
9.92611 

8 
8 
8 
8 

35 
34 
33 
32 
3i 

6 

8 

20 

2.6 

2.4 

2.? 

20 

2.0 

2-3 
2-6 

19 

I.§ 

2-3 
2.6 

30 

3i 

32 
33 
34 

9.73021 
9.73041 
9.73o6i 
9.73081 
9-73  ioi 

19 
20 
2O 

19 
20 

9.80413 
9.80440* 
9.80474: 
9.80502 
9.80  530 

27 

28 
28 
28 

2? 
«0 

0.19  581 

0.19553 
0.19525 
0.19497 
o.  19470 

9.92603 
9.92  595 
9.92  587 
9.92  579 
9.92  576 

8 
8 
8 
8 

30 

29 
28 
27 
26 

9 

10 

20 
30 
40 

3-1 

3-4 
6-8 

10.2 
13-6 

3-0 

3.3 

6.6 

10.0 

13-3 

ff.  2 

% 

6.5 

9-? 
13.0 

jf.    G 

P 

37 
38 
39 

9.73120 
9.73146 
9-73  1  60 
9.73  1  80 
9-73I99 

19 

20 

19 

20 

19 

9.80558 
9.80586 
9.80613 
9.  80  641 
9.80669 

28 

2? 
28 
28 

~Cj 

0.19442 
0.19414 
0.19386 

0.19358 
0.19336 

9.92  562 
9.92554 
9-92  546 
9-92  538 
9.92  530 

8 
8 
8 
8 

25 

24 
23 

22 
21 

5° 

I7.I 

10.6 

10.2 

40 

4i 

42 
43 
44 

9.73219 
9.73239 
9-73  258 
9-73278 
9.73298 

19 
19 

20 

19 

9.  80  697 
9.80725 
9.80752 
9.80786 
9.80803 

27 
28 
2? 
28 
28 
~9 

0.19303 
0.19275 
0.19  24? 
0.19  219 
0.19  191 

9.92  522 
9.92514 
9.92  506 
9.92498 
9.92489 

8 
8 
8 
8 

20 

19 
18 

17 
16 

f 

8 

>    o.§ 

8 

0.8 

? 

o.? 

45 
46 

47 
48 

49 

9-7331? 
9-7333? 
9-73357 
973376 
9  73396 

19 

20 

19 
If 

i§ 

9.80836 
9.80864 
9.80891 
9.80919 
9.80947 

27 
28 
2? 
28 

2? 

_0 

0.19  164 
0.19  136 
0.19  io§ 
0.19086 
0.19053 

9.92481 
9-92473 
9-92465 
9.92457 
9.92449 

8 
8 

8 
8 

15 

14 
13 

12 
II 

; 

c 

1C 

2C 

'     I.O 

}  1.  1 

>    i-3 
>    1.4 

>  ^-8 

0.9 
1.6 

1.2 

i-3 

2-6 

0.9 
I.O 

I.I 

1.2 

2-5 

50 

5i 
52 
53 
54 

9-734I5 
9-73435 
9-73455 
9-73474 
9-73494 

19 

20 

I§ 
19 

19 

9.80975 

9.8l  002 
9.8l  030 
9.8l  058 
9.8l  OS^ 

25 

2? 
2? 
28 

2? 
-o 

0.19025 
o.  1  8  99? 
o.i  8  970 
o.  1  8  942 
0.18914 

9.92441 

9-92433 
9.92424 

9.92416 
9.92403 

8 
8 

8 
8 
8 

10 

9 
8 

6 

3^ 
4^ 
5^ 

>   4-2 
>    5-6 
>   7.i 

4.0 

11 

3-? 
5-o 

6.2 

!! 
S 

59 

9-735I3 
9-73533 
9-73  552 
9-73  572 
9-73  59i 

19 

'? 

19 
19 
i§ 

9.8l  113 
9.8l   141 

.9.81  i6g 
9.81  195 
9.81  224 

25 

2? 
2? 

28 

2? 

0.18886 
0.18859 
0.1883! 
0.18803 
o.  1  8  776 

9.92400 
9.92392 
9-92  383 
9-92  375 
9-9236? 

8 

8 
8 
8 

5 
4 
3 

2 

I    60 

9.73611 

19 

9.81  251 

2  7 

0.18748 

9-92359 

8 

0 

Log.  Cos. 

d. 

Log.  Cot. 

\  C.  (1. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

p. 

p. 

57 


.  380 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

33 


t 

Log.  Sin. 

d. 

Log.  Tan. 

r.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

F 

.  P. 

0 

I 

2  " 

3 

4 

9.73611 
9.73  636 
9.73650 
9.73669 
9.73688 

19 
19 
19 
19 

TO 

9.81  251 
9.8l  279 
9.81  307 

9-8i  334 
9.81  362 

28 
2? 
2? 
28 

~B 

0.18743 
o.i  8  726 
0.18693 
6.18665 
0.1863? 

9-92359 
9.92351 
9.92  342 
9.92334 
9-92  326 

8 

8 
8 
8 
8 

60 

9 

I 

6 

8 
9 

9.73708 
9-7372? 

9-73746 
9  73  766 

973785 

19 

19 
19 

-19 
if 

TO 

9.81  390 
9.8141? 
9.81445 

9.8i473 
9.81  500 

27 

2? 

i 

2? 

~a 

0.18610 
0.18  582 
0.18555 
o.i  8  527 
o.  1  8  499 

9.92318 
9.92310 

9-92  301 
9.92293 
9.92285 

o 

8 

8 
8 

8 

3 

55 
54 
53 
52 
5i 

28 

6     28 

2? 

2  f 

27 

2  7 

10 

ii 

12 
13 
U 

9.73805 
9-73824 

9.73843 
9.73862 
9.73882 

I9 
19 
19 
19 
19 

in 

9.81  528 
9.81555 
9.8i  583 
9.81  616 
9.81  638 

27 
2? 
2? 
2? 
2? 

_o 

0.18472 
0.18444 
0.18417 
0.18389 
o.i  8  362 

9.92277 
9-92  263 
9.92  266 
9.92252 
9.92  244 

8 
8 

8 
8 

50 

49 
48 

47 
46 

7     3-2 
8     3-? 
9    4-2 
10    4.5 

20      9.3 

3-2 

3-6 
4.1 
4.6 
Q.I 

a 

4.0 

4.5 
9.0 

II 

17 

18 
19 

9.73901 
9.73926 
9-73940 
9-73959 
9-73978 

19 
19 
19 
«9 

19 

9.81  666 
9-8i  693 
9.81  721 
9.81  748 
9.81  776 

25 
2? 
2? 
2? 

2? 

a 

0.18334 
0.18305 
o.  18  279 
0.18251 
o.i  8  224 

9.92235 
9.92  227 
9.92219 

9.92  210 
9.92  2O2 

. 

8 
8 

45 
44 
43 
42 

4i 

30    14.0 

40  i8.g 
50  23.3 

I3-? 
18.3 
22.9 

13-5 

1  8.0 

22.5 

20 

21 

22 

23 
24 

9-7399? 
9.74015 
9.74036 
9.74055 
9.74074 

I9 

\l 

19 
19 

in 

9.81  803 
9.81  831 
9.81  853 
9.81  886 
9.81913 

2? 
2? 
2? 
2? 

2? 

Oo 

0.18  195 
o.  18  169 
0.18  141 
o.i  8  114 
0.18085 

9.92  194 
9.92  185 

9.92  17? 

9.Q2  169 

9.92  166 

8 

8 
c 

40 

38 

g 

25 
26 
27 
28 
29 

9-74093 
9.74112 

9-74  I31 
9-74I51 
9.74170 

19 

19 
19 
19 
19 

9.81  941 
9.81  963 
9.81996 
9.82023 
9.82051 

27 
2? 
2? 
2? 

2? 
_/s 

0.18059 
0.18031 
o.  1  8  004 
0.17975 
0.17949 

9.92152 

9.92  144 
9.92  135 
9.9212? 
9.92  119 

8 
8 

5 

35 
34 
33 
32 
3i 

19 

6     1.9 
7     2.3 
8     2.6 

19 

1.9 
2.2 
2-5 

18 

1-8 

2.1 

2.4 

-    Q 

30 

3i 
32 
33 
34 

9.74189 
9.74208 
9-74227 
9.74246 
9.74265 

I9 

1Q 

*9 
19 

19 

9.82073 
9.82  io§ 
9.82  133 
9.82  166 
9.82  1  88 

27 
27 
2? 
2? 

2? 

e 

0.17  921 
0.17  894 
0.17  867 
0.17839 
0.17  812 

9.92  no 

9.92  IO2 
9.92094 
9.92085 
9.92077 

8 

8 

8 

30 

29 
28 

27 
26 

9     2.9 
10     3.2 

20      6.5 

30     9-? 
4°   13-° 
50   1  6  2 

2-8 
3-1 
6.3 
9.5 

12.5 
ic  Q 

3'I 

6.! 

9-? 
12.3 

I  S.A. 

P 

37 
38 
39 

9.74284 

9.74303 
9.74322 

9-74341 
9.74360 

19 
i§ 
19 
19 
18 

9.82215 
9.82  243 
9.82  276 
9.8229? 
9.82  325 

27 

2? 
2? 
27 
2? 

0.17784 
0.17757 
0.17729 
0.17  702 

0.17675 

9.92069 
9.92066 
9.92052 
9.92043 
9.92035 

8 

8 

25 
24 

23 

22 
21 

40 

41 

42 

43 
44 

9-74379 
9-74398 
9.74417 
9-74436 
9-74455 

19 
19 
19 
19 
J9 

9.82352 
9.82380 
9.82407 
9.82434 
9.82  462 

2? 

2? 
27 
2? 
2? 

0.1764? 
o.  1  7  620 
0.17593 
0.17565 
0.17538 

9.92027 
9.92018 

9.92  oio 
9.92001 
9-91  993 

1 

8 

5 

20 

19 

17 
16 

6 

8      8 

3.3   o. 

8 

4I 
46 

47 
48 

49 

9-74474 
9-74493 
9-7451? 
9-74530 
9-74549 

I9 
19 
i§ 
19 
19 

9.82489 

9-82516 
9.82  544 
982571 
9.82  593 

27 

27 

2? 

2? 

27 
_/5 

0.17  510 
0.17483 
0.17456 
0.17423 
0.17401 

9.91  984 
9.91  976 
9.91  96? 
9.91  959 
9.91  951 

8 
8 
8 
8 

8 

5 

15 
H 
13 

12 
II 

I 

9 

10 
20 

1.0    0. 

I.I    I. 

1.3  I. 
1.4  I. 

2.8     2. 

9 
6 

2 
I 

50 

51 

52 
53 
1     54 

9-  74568 
9.74587 
9.74606 
9-74625 
9-74643 

19 
i§ 
19 
19 
18 

9.82626 
9.82653 
9.82686 
9.82708 
9.82735 

2? 
27 
2? 
2? 
27 

0.17374 
0.17347 
0.17319 
0.17  292 
0.17  265 

9.91  942 

9.91  934 
9.91  925 
9.91917 
9.91  903 

8 

i 

8 

10 

9 
8 

6 

30    < 
40 
50 

\.'2     4. 

5-6    5- 
7.1    6. 

0 
1 

III 
H 

59 

9.74662 
9.74681 
9-74700 
9-74718 
9-7473? 

^ 

18 

19 

18 
19 

9.82762 
9.82789 
9.82817 
9.82  844 
9.82871 

2? 

27 
2? 

27 
2? 

0.1723? 

O.I7  210 
O.I7  183 
0.17  156 

0.17  I2§ 

9.91  900 
9.91  891 
9.91  883 
9.91  874 
9.91  866 

8 

1 
8 

5 
4 
3 

2 

I 

(>0 

9-74756 

18 

9  82  893 

27 

0.17  101 

9.91  85? 

8 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

1 

'.  P. 

56 


381 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

34° 


t 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p. 

P. 

jl 

0 

I 

2 

3 

4 

9.74756 

9-74775 
9-74793 
9.74812 

9-7483I 

19 

18 
19 
18 

9.8289§ 
9.82  926 
9.82953 
9.82986 
9.8300? 

2? 
27 
2? 

27 
~a 

0.17  ioi 

0.17074 
0.17047 
0.17019 
0.16992 

9.9I  85? 
9.91  849 
9.91  846 
9.91  832 
9.9I  823 

8 
8 
8 
8 

60 

59 
58 

i 

I 

8 

9 

9.74849 
9.74863 
9.74887 

9.74905 
9.74924: 

19 
19 

9.83035 
9.83062 
9.  83  089 
9.83115 
9-83  H3 

27 
27 
2? 

27 

27 
Oa 

0.16965 
0.16938 
0.16916 

0.16883 
0.16855 

9.91  814 

9.91  806 

9.91  79? 
9.91  789 
9.91  786 

8 
8 
8 
8 

55 
54 
53 
52 

2? 

6       2  9 

27 

2  7 

2*6 

10 

ii 

12 
13 
H 

9-74943 
9.74961 
9.74980 

9-74998 
9.75017 

18 
'8 

9.83171 
9.83  I98 
9.83225 
9.83252 
9.83279 

27 
27 
2? 

27 
27 

-79 

0.16829 
o.i  6  802 
0.16774 

0.1674? 
o.  16  726 

9.91  772 
9.91  763 

9-91  755 
9.91  746 

9-91  73? 

1 

8 
9 
8 

8~ 

50 

49 

48 
47 
46 

1  11 

9     4-1 
10     4.6 

20      9.1 

3-6 
4.6 

4-5 
q.O 

3-1 

3-5 
4.0 

4-4 

8-8 

15 

16 

17 
18 

19 

9.75036 
9.75054 
9-75073 
9-75  091 
9-75  no 

19 
18 
18 
18 
18 

9-83307 
9-83334 
9.83361 
9.83388 
9.83415 

27 
27 
27 

27 
2? 

o.  1  6  693 
0.16666 
0.16639 
o.  16  612 
0.16  584 

9.91  729 
9.91  726 
9.91712 
9.91  703 
9.91  694 

8 
8 

45 
44 
43 
42 

30    I3-? 
40    18.3 
50    22-9 

13-5 
18.0 
22.5 

13-2    , 
17-6 

22.1 

20 

21 
22 

23 

i      24 

9-75  I28 
9-75  H7 

9.75  165 
9-75  184 
9.75  202 

18 
18 
18 
18 

9.83442 
9.83469 

9-83496 
9.83524 

9-83551 

27 

27 
27 

2? 
27 

0.1655? 
o.i  6  536 
o.i  6  503 
0.16476 
0.16449 

9.91  686 
9.91  67? 
9.91  66§ 
9.91  660 
9.91  651 

8 
9 
8 
8 

40 

39 
38 

25 

26 
27 
28 
29 

9.75221 

9-75239 
9.7525? 
9.75276 
9.75294 

18 
'§ 

51 

9-83  578 
9-83605 
9.83632 
9.83659 
9-83685 

27 
27 
27 
2? 
27 

0.16422 
0.16395 
0.16368 
o.i  6  346 
0.16  313 

9.91  642 
9.91  634 
9.91  625 
9.91615 
9.91  608 

8 

8 

5 

35 

34 
33 
32 

19 

6     1.9 

7       2.2 

8     2.5 

18 

!•£ 
2.1 
2.4 
i  8 

18 

1.8 

2.1 

2-4 

0     1 

30 

32 
33 
1    34 

9.75313 
9.75331 
9-75349 
9.75368 
9-75  386 

18 
'§ 

9.83713 
9.83740 
9.8376? 

9-83794 
9.83  821 

27 
27 
27 
27 
27 

o.i  6  285 
o.  16  259 
o.  16  232 
o.i  6  205 
o.  16  178 

9.91  599 
9.91  596 
9.91  582 

9-91  573 
9.91  564 

8 

1 

8 
9 

Q 

30 

29 
28 

9     2-8 
10     3.1 

20      6.3 

30     9-5 
40   12.5 

co    I  C  § 

6J 

9.2 
12.2 

I  $.4 

0 

6.0 
9.0 

12.  0 
I  5.O 

I    35 
36 

39 

9-75404 

9.75441 

9-75459 
9.75478 

18 
18 

'i 

T  8 

9.83848 
9.83875 
9.83902 
9.83929 
9-83957 

27 
27 

27 

27 
2? 

o.  16  151 
o.  16  124 
0.1609? 
0.16076 
0.16043 

9.91  556 
9-9i  547 
9-91  538 
9.91  529 
9.91  521 

8 
9 
8 
9 
8 

25 

24 

23 

22 
21 

40 

42 
43 
44 

9.75496 
9-75  5H 
9-75  532 
9-75551 
9-75  569 

15 

18 

9-83984 
9.84011 
9.84038 
9.84065 
9.84091 

27 
27 
27 
27 
26 

0.16016 
0.15989 
o.  1  5  962 
0.15935 
0.15908 

9.91  512 
9.91  503 
9.91  495 
9.91  486 
9.91  47? 

9 
8 
9 
8 

20 

19 

18 

17 
16 

6 

9 

0.9  c 

§ 

>-8 

46 
47 

48 

49 

9-75  587 
9.75  605 
9.75623 
9.75642 
9.75660 

18 

18 
18 

18 
18 

T   Q 

9.84113 
9-84  H5 
9.84  172 
9.84  199 
9.84225 

27 
27 

27 
27 

27 

0.15  88T 
0.15854 
0.1582? 
0.15806 
0.15  773 

9.91  463 
9.91  460 
9.91451 
9.91  442 
9-91  433 

9 

8 
9 
9 
8 

15 
13 

12 
II 

9 

10 
20 

I.O     ] 
1.2     1 

i.S    i 
1.5    i 

3-0   : 

.0 

.1 

-3 
•  4 
J 

50 

52 
53 
54 

9.75678 
9-75696 
9-75  7H 
9-75732 
9.75756 

15 

18 
18 
18 
18 

9.84253 
9.84286 
9.8430? 

9-84334 
9.84  361 

27 
27 

27 

27 

26 

o.  1  5  745 
0.15719 
o.  1  5  692 
0.15665 
o.  1  5  639 

9.91  424 
9.91  416 
9.91  407 
9.91  398 

9 
8 
9 

1 

10 

9 
8 

6 

30 
40 
50 

J'b   ' 
6.0    « 

7.5  ; 

k* 

ii 

P 

57 
58 
59 

9.75769 
9.75787 
9-75  805 
9-75823 
9.75841 

18 
18 
18 
18 

9.84388 
9.84415 
9.84442 
9.84469 
9.84496 

27 

27 

27 
27 

27 

0.15  612 
0.15  585 
0.15558 

0.15504 

9.91  386 
9.91  372 
9.91  363 
9-91  354 
9-91  345 

8 
9 
9 
8 

5 
4 
3 

2 

60 

9.75859 

Io 

9.84522 

2S 

0.1547? 

9-91  336 

9 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.d. 

Log.  Tan. 

Log.  Sin. 

d. 

f 

1 

\  p. 

55 


382 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

35° 


t 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p. 

P. 

0 

I 

2 

3 
4 

9.75  877 
9.75895 
9.759I3 
9-75931 

18 
18 
18 
18 

To 

9.84522 

9.84  549 

9-84  576 
9.84603 
9.84630 

27 
27 
27 
26 

0.1547? 

o.  1  5  456 

0.15423 
0.15396 

o.  1  5  370 

9-91  336 
9.91  327 

9-91  3*8 
9.91  310 
9.91  301 

9 
9 
8 
9 

60 

p 
1 

1 

1      8 
)      9 

9-75949 
9.75967 

9.75985 
9.76003 

9.  76  02  1 

18 
18 
18 

18 

9.84657 
9.84684 
$.84711 
9-8473? 
9.84764 

27 

27 
27 

26 

27 

0.15343 
0.15  316 
o.  1  5  289 
o.  1  5  262 
0.15235 

9.91  292 
9.91  283 
9.91  274 
9.91  265 
9.91  256 

9 
8 
9 
9 
9 

55 
54 
53 
52 

27 

6,, 

28 

_  ? 

10 

n 

12 
13 
14 

9.76039 
9.76057 
9.76075 
9.76092 

9.76  no 

18 
18 
18 

18 

To 

9.84791 
9.84818 
9.84845 
9.84871 
9.84893 

27 

26 

27 

26 

27 

0.15208 
0.15  182 
0.15155 
o.  1  5  1  2§ 
0.15  lot 

9.91  247 
9.91  239 
9.91  230 

9.91  221 
9.91  212 

9 
8 
9 
9 
9 

50 
49 
48 
47 
46 

2. 

I     I 
9     4. 
10     4. 

2O       Q 

7 

6 
6 

2.5 

3.1 

3-5 
4.0 

4.4 

8  8 

II 

17 

18 
19 

9.76123 
9.76146 
9.76  164 
9.76  182 

9.  76  200 

lo 

18 

Is 

18 

T  G 

9.84925 
9.84952 
9.84979 
9.85005 
9-85032 

27 
26 

27 

26 

27 

0.15074 
o.  1  5  048 

0  15  021 

0.14994 

o.  14  96? 

9.91  203 
9.91  194 

9.91  i8§ 
9.91  176 
9.91  1  6? 

9 
8 
9 
9 

45 
44 
43 

42 

***     y- 

30   13 
40    1  8. 

50    22. 

5    i 

0     I 

5   2 

3.2 

7.6 

2.1 

20 

21 

22 
23 

24 

9.7621? 
9.76235 
9.76253 
9.76271 
9.76289 

17 

18 
18 

18 

9.85059 
9.85086 

9-85113 
9.85  139 

9.85  166 

27 
26 

27 
26 

27 

0.14940 
0.14914 
o.  14  887 
0.14866 
0.14833 

9-91  J58 
9.91  149 
9.91  146 
9.91  131 

9.91  122 

9 
9 
9 
9 
9 

40 

39 
38 

I 

25 
26 

27 
28 
29 

9.76306 
9.76324 
9.76342 
9.76  360 
9.7637? 

18 

18 

i? 

-0 

9-85  193 

9.85  220 
9.85246 
9.85  273 
9.85  300 

27 
26 
27 
26 

0.14807 
0.14780 

0.14753 
0.14726 

o.  14  700 

9.91  113 
9.91  104 
9.91  095 
9.91  086 
9.91  07? 

9 
9 
9 
9 

35 
34 
33 
32 

18 

6     1.8 
7     2.1 
8     2.4 

I? 

j   ; 
2.C 

2.; 

17 
r       1.7    i 
)       2.0 

J       2.2 

30 

32 
33 
34 

9.76395 
9.76413 
9.76431 

9-76448 
9.  76  466 

lo 
18 

T  Q 

9.85327 

9-85353 
9.85380 
9.85407 
9-85433 

27 
26 
26 

27 

26 

0.14673 
0.14646 
o.  14620 

0.14593 
0.14566 

9.91  06§ 
9.91  059 
9.91  056 
9.91  041 
9.91  032 

9 
9 
9 
9 
9 

30 

29 
28 

27 
26 

9     2.7 
10     3.0 

20       6.0 

30    9.0 

4O     I2.O 

2.t 
2.C 

S-1 

«.J 
n.  ( 

)       2.5    1 

)   2.3  ; 

I     5.6 

r       8.5 

>    "'3   i 
H* 

35 
36 

39 

9.76484 
9.76501 
9.76519 
9-76536 
9.76554 

18 

I? 
I? 
I? 

18 

9.85466 
9.85487 
9.855I3 
9.85  546 

26 
26 

27 
26 

0.14539 
0.14513 

0.14486 
0.14459 

0.14433 

9.91  023 
9.91  014 
9.91  005 
9.90996 
9.9098? 

9 
9 
9 
9 
9 

A 

25 
24 
23 

22 
21 

•C 

•! 

40 

42 
43 
44 

9-76572 

9-7658§ 
9.  76  607 
9.76624 
9.76642 

il 

9.85  594 
9.85626 
9-85647 
9-85673 
9.85706 

26 

26 

26 

27 

0.14406 
o.  14  379 

0-14353 
0.14326 
0.14299 

9.90978 
9.90969 
9.90960 
9.90951 
9.90942 

9 
9 
9 
9 
9 

20 

19 
18 

17 
16 

9 

6   0.9 

9 

O.q 

8 

0.8 

3 

47 
48 

49 

9.76660 
9.7667? 
9.76695 
9.76712 
9.76730 

i? 
i? 

T  G 

9.85727 
9.85  753 
9.85786 

9.85807 
9-85833 

26 
27 

26 
26 

0.14273 
o.  14  246 
0.14  219 
0.14193 
0.14  1  66 

9-90933 
9.90923 
9.90914 
9.90905 
9.90896 

9 
'9 
9 
9 
9 

15 
13 

12 
II 

7    LI 

8     1.2 

9    1.4 
10    1.6 

20    3-T 

I.O 
1.2 
1-3 

3-o 

I.O 

i.i 

1.4 
2-8 

50 

52 
53 

54 

9.76747 
9.76765 
9.76782 
9.76800 
9.7681? 

i; 

I? 
i? 
i? 

T  fr 

9.85  860 
9.85887 
9.85913 
9.85940 
9-85966 

27 

26 
26 
26 

0.14  140 
0.14  113 
o.  14  086 
0.14060 
0.14033 

9.9088? 
9.90878 
9-  90  869 
9.  90  860 
9.90856 

9 
9 
9 

10 

6 

30   4-? 
40   6.3 
50   7.9 

4-5 
6.0 

7-5 

4-2 
5-6 

59 

9.76835 
9.76852 
9.76869 
9.76887 
9.76904 

17 

17 

9.85993 
9.  86  020 
9.86046 
9.86073 
9.86099 

27 

26 
26 
26 

0.14007 
0.13980 

0.13953 
0.13927 
0.13906 

9.90  841 
9.90832 
9.90823 
9.90814 
9.90805 

y 
9 
? 
9 
9 

5 
4 
3 

2 

60 

9.76922 

A/ 

9.86  126 

25 

0.13874 

9.90796 

y 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

' 

p. 

p. 

383 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

36° 


1    , 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p 

.  P. 

0 

I 

2 

3 
4 

9.76922 
9.76939 

9.76974 
9-7699I 

If 
17 

9.86  126 
9.86  152 
9.86  179 
9.  86  206 
9.86232 

28 
28 

27 

26 

0.13874 
0.1384? 
0.13  821 
0.13794 
0.1376? 

9.90796 

9.90788 
9.9077? 
9.90768 
9.90759 

9 
9 
9 
9 

60 

59 
58 

% 

I 

8 
9 

9-77008 
9.77026 
9-77043 
9.77066 
9.77078 

17 

If 
If 
17 
If 

9.86  259 
9.8628$ 
9.86  312 

9-86338 
9.86365 

26 

28 

28 
25 

0.13741 
0.13714 
0.13688 

o.  1  3  66! 
0.13635 

9.90750 
9.90746 
990731 
9.90722 
9.90713 

9 
9 
9 

9 
9 

55 

54 
53 
52 

27 

26 

-7    Z 

26 

10 

ii 

12 

13 

14 

9.77095 
9.77  112 
9.77130 
9-77  147 
9-77  164 

17 
If 
If 
17 
If 

9.86  391 
9.86418 
9.86444 
9.86471 

26 

28 

28 
28 

~? 

0.13608 
0.13  582 

0.13529 
0.13502 

9.90703 

9.90694 
9.90685 
9.90676 
9.90665 

9 
9 
9 
9 

50 

49 
48 

47 
46 

2.7 

7     3-i 
8     3.6 
9     4-0 
10     4.5 
20     q.o 

2.5 

3-i 

3-$ 
4.0 
4.4 
8.A 

3-o 
3-4 
3-? 
4-3 
8.8 

;i 

17 

18 
19 

9-77  181 
9-77  198 
9.77216 

9-77233 
9.77256 

17 
17 
if 
17 
If 

9.86  524 
9.86556 

9.86577 
9.86603 
9.86630 

26 
28 
28 
28 
28 

~2 

0.13476 
0.13449 
0.13423 

0.13398 
0.13370 

9.9065? 
9.90648 
9.90639 
9.  90  629 
9.  90  626 

9 
9 
9 
9 
9 

45 
44 
43 
42 

30   13.5 
40   1  8.0 
50  22.5 

13.2 
17-8 

22.1 

13.0 
17.3 
21.6 

20 

21 

22 

23 
I      24 

9.7726? 
9.77284 
9-77  302 
9-773I9 
9-77336 

17 
17 
If 
17 
17 

T£ 

9.86655 
9.86683 
9.86709 
9.86736 
9.86762 

26 
28 
28 
28 

26 
~< 

0.13343 
0.13317 
o.  1  3  296 
o.  1  3  264 
0.1323? 

9.90611 
9.90  602 
9.90592 
9.90583 

9.90  574 

9 
9 
9 
9 
9 

40 

39 
38 

11 

25 
26 
27 
28 
29 

9-77353 
9.77376 

9-77  38f 
9.77404: 
9.77421 

17 
17 
17 
17 
17 

T? 

9-86783 
9.86815 
9.86841 
9.86868 
9.86894 

26 

28 

26 

28 
~? 

0.  1  3  2  1  1 

0.13  185 

0.13158 
0.13132 

0.13  10$ 

9.90  564 

9.90  555 
9.90  546 

9-90538 
9.90527 

9 
9 
9 
9 
9 

35 
34 
33 
32 

6     i.? 
7     2.6 
8     2.3 

17 

1-7 
2.0 

2.2 

16 

i-8 

1-9 

2.2 

30 

32 
33 
34 

9-77439 
9.77456 

9-77473 
9-77490 
9-77  5°7 

1/ 
17 
17 
17 
17 

9.86921 
9-8694f 
9.86973 
9.  87  ooo 

9.87026 

26 
28 
26 

28 
28 

0.13079 

0.13052 

0.13025 

o.  1  3  ooo 
0.12973 

9.90518 

9-90  508 
9.90499 
9.90490 
9.  90  486 

9 
9 
9 
9 
9 

30 

29 
28 
27 
26 

9     2.6 
10     2.9 

20       5.8 

30     8.? 
40   n.g 

2-$ 

2-8 

5-8 

II.  § 

T/1    T 

2.5 

2.? 

5-5 

3.2 
II.  0 

35 
36 

39 

9-77  524 
9-77  54i 
9-77  558 
9-77  575 
9-77  592 

17 
17 
17 
17 

9.87053 
9.87079 

9.87  10$ 

9.87  132 
9.87158 

28 
28 

">2 

0.12947 

0.  1  2  926 
0.12  894 

o.  1  2  868 

0.12  841 

9.90471 
9.  90  461 
9.90452 
9-90443 
9-90433 

9 
9 
9 
9 

25 

24 

'23 

22 
21 

14.  i 

*3-7 

40 

42 
43 
44 

9.77609 
9.77626 

9-77643 
9.77  660 
9.77677 

17 
17 
17 
17 
17 

9-87  185 

9.87  211 

9-8723f 
9.87  264 
9.87296 

26 

26 

28 
28 
28 

_/r 

0.12  815 
0.12  789 
0.  12  762 
0.12  736 
0.12  709 

9.90424 
9.90414 
9.9040$ 
9.90396 
9.90385 

9 
9 
9 
9 
9 

20 

19 
18 

17 
16 

6  < 

9      9 

D.§    0. 

9 

'46 

47 
48 

49 

9.77693 
9.77716 
9.7772? 

9-77744 
9.77761 

17 
17 

17 

9-87318 
9-87343 
9.87369 

9.87422 

2O 
28 

28 
26 

28 

0.12683 
0.12  657 
0.12  636 
0.  1  2  604 
0.12  578 

9-90377 
9.90  36? 
9.90358 
9-90348 
9-90339 

9 
9 

I 

15 
14 
13 

12 
II 

8 
9 

10 

20 

i.i    i. 

Si; 

3-i    3- 

6 

2 

3 
5 

0 

50 

52 
53 
54 

9.77778 

9-77795 
9.77812 

9.77823 

17 
17 

17 

9.87443 

9.87474 
9.87501 
9.87  52f 

9-87553 

2b 
26 

28 
28 

O.I255I 
0.!252$ 

o.  1  2  499 

0.12472 
0.12445 

9.90330 
9-90326 
9.90311 
9.9030! 
9.90292 

9 
9 
9 
9. 

10 

7 
6 

30    ' 
4O    ( 

5° 

*-f   4- 
5-3   6. 
7-9    7- 

5 

0 

5 

P 

ill 

59 

9.77862 
9.77879 
9.77896 

9-779I3 
9.77929 

17 

18 
17 

17 

9.87  580 
9-87605 
9.87632 
9.87659 
9.87685 

26 
26 

O.  I  2  42O 
0.12393 
0.  1  2  36? 
O.I234I 
O.I23I5 

9.90  282 
9.90  273 
9.90  263 
9-90254 
9.90244 

9 
9 
9 
9 
9 

5 
4 
3 

2 
I 

60 

9-77946 

17 

9.87711 

25 

0.12288 

9  90235 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

' 

p 

.  P. 

53 


3»4 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

37° 


r      ! 

LOR.  Sin. 

d. 

I.IIL'.  Tan. 

r.  d. 

Log.  Cot. 

Log.  Cos.  i 

d. 

P.  V. 

0 

I 

2 

3 
4 

9-77946 
9.77963 

9-77  98° 
9-77996 
9.78013 

16 
17 
16 
17 

tZ 

9.8771! 

9.8773? 
9.87764 
9.87796 
9.87816 

26 

26 
26 

26 

?2 

O.I228§ 
0.12  262 
0.12  236 
0.  1  2  20§ 
0.12  183 

9.90235 
9.90225 
9.90216 

9.90  205 

9.90  196 

1 

9 

10 
A 

00 

59 
58 
57 
56 

6 

8 
9 

9.78030 
9-78046 
978063 
9.78080 
9.78097 

16 
16 
17 
16 

17 
~ 

9.87843 
9.87869 

9^7  895 
9.87921 
9.87948 

26 

26 

26 

26 

26 
•76 

0.12  157 
0.12  131 
0.12  104 
0.12078 
0.12  052 

9.90187 
9.90  17? 

9.90  1  68 
9.90153 
9.90  149 

9 
9 

§ 

n 

55 
54 
53 
52 
5i 

2g 

6-7    2 

26 

*j  f\ 

10 

ii 

12 

13 
14 

9.78113 
9.78130 
9.78  147 
9.78  163 
9.78  1  80 

*6 
16 

17 

4 

16 
rp 

9.87974 
9.88006 
9.88025 
9.88053 
9.88079 

26 

26 

26 
26 

?6 

0.  1  2  026 

o.i  i  999 
o.  1  1  973 
o.i  i  947 
o.  ii  921 

9.90139 
9.90  130 

9.90  120 

9.90  i  16 
9.90  101 

9 
9 

10 

9 
9 
5 

50 

49 
48 
47 
46 

2-6 

7     3.1 
8     3-5 
9     4.0 
10     4.4 
20     8.§ 

3° 
3-4 
3-9 
4-3 

8.6 

!i 

17 

18 
19 

9.78  196 
9.78213 
9.78  230 
9-78  246 
9.78263 

*6 
*6 
17 
16 
16 

9.88  105 
9.88131 
9-8815? 
9.88  184 
9.88210 

26 
26 

8 

?< 

o.  ii  895 
o.  1  1  86§ 
o.  1  1  842 
o.  ii  816 
o.  1  1  790 

9.90091 
9.00082 
9.00072 
9.90062 
9.90053 

y 
9 

10 

9 
9 

45 
44 
43 
42 
4i 

30   13.2 
40   17.6 

50    22.1 

13.0 
17.3 
21.6 

20 

21 
22 

23 
24 

9.78279 
9.78  296 
9.78312 
9.78329 
9.78  346 

*6 
'6 

16 
17 

r2 

9-88236 
9.88262 
9.88283 
9.88315 
9.88341 

26 
26 
26 

3 

26 

o.  1  1  763 
o.i  i  73? 
o.  1  1  711 
o.i  i  685 
o.  ii  659 

9.90043 
9.90033 
9.90024 
9.90014 
9.90004 

10 

9 
9 

10 

9** 

40 

39 
38 

% 

25 
26 

27 
28 
29 

978362 
9-78379 
9-78395 
9.78412 
9.78428 

16 
16 
16 

J£ 

16 

Tp 

9.88367 
9.88393 
9.88419 

9-88445 
9.88472 

9 

26 
26 

?6 

o.  1  1  633 
o.  1  1  605 
o.  1  1  586 
o.i  i  554 
o.  ii  528 

9.89995 
9.89985 

9.89975 
9.89966 

9-89956 

9 

10 

9 

9 

35 
34 
33 
32 
3i 

17     ie 

6     1.7   -i. 
7     2.0     i. 

8       2.2       2. 

;    16 

6      1.6  ! 
9      1-8 

2        2.1 

30 

3i 

32 
33 
!    34 

9.78444 
9.78461 

9-7847? 
9.78494 
9.78516 

J6 
16 
16 

ii 

T  2 

9.88498 
9.88  524 
9.88556 
9-88  576 
9.88602 

26 

26 
26 
26 
?2 

o.  ii  502 
o.  1  1  476 
o.  1  1  449 
o.  1  1  423 
o.i  i  397 

9.89946 
9.89937 
9-8992? 
9.8991? 
9.  89  908 

9 
9 

10 

9 

30 

29 
28 
27 
26 

9     2.5     2. 

10       2-8       2. 

20     5.6     5- 
30     8.5     8. 

40  ii.  §  ii. 

5     2.4  ! 
?    2.6 
5     5-3 

2        8.0 
0     10.6     | 
5      \1\ 

3 

37 
38 
39 

9.78  527 
9-78  543 
9-78  559 
9.78576 
9-78  592 

16 
16 
16 

16 

16 
_  p 

9.88629 
9.88655 
9.88681 
9.88707 
9-88733 

26 
26 
26 
26 

26 
oft 

o.i  i  371 
o.  1  1  345 
o.n  319 
o.  ii  293 
o.  1  1  265 

9.89898 
9.89883 
9.89873 
9.89  869 
9.89859 

9 

10 

9 

10 

25 

24 

23 

22 
21 

5°    *4-  l    13- 

7    13o 

40 

4i 
42 
43 
44 

9.78609 
9.78625 
9.78641 
9.78658 
9.78674 

J6 
16 

JS 

16 
~ 

9.88759 
9.88785 
9.88811 
9.88838 
9.88864 

26 
26 

26 
26 
0/r 

o.  ii  240 
o.  ii  214 
o.i  i  183 
o.  ii  162 
o.  ii  136 

9.89849 

9-89839 
9.89830 
9.89820 
9.89816 

9 

IO 

9 

10 

9 

20 

19 
18 

17 
16 

10 

6     .0 

9 

0.9 

45 
46 
47 
48 
49 

9.78696 
9.78707 
9.78723 
9.78739 
9-78755 

!6 

16 
16 

II 

~ 

9.88890 
9.88916 
9.88  942 
9.88968 
9.88  994 

20 

26 
26 
26 
26 

-/• 

o.  1  1  no 
o.  1  1  084 
o.  n  058 
o.  n  032 

0.  1  1  005 

9.89806 
9.89791 
9.89781 

9.89771 
9.89761 

9 

10 
10 

9 

15 

14 
13 

12 
II 

7     -I 
8     -3 
9     -5 
10     .6 
20   3.3 

i.i 

1.2 
1.4 

1.6 
3-1 

50 

5i 
52 
53 

i     54 

9.78772 
9.78783 
9.7880$ 
9.78821 
9-78837 

ib 
16 
16 

1 

16 

>r 

9.89  020 
9.89045 
9.89072 
9.89098 
9.89  124 

2O 
26 
26 
26 
26 
(• 

o.  10  979 
0.10953 
0.1092? 
o.  10  901 
0.1087$ 

9.89751 
9-89742 
9.89732 
9.89722 
9.89712 

§ 

IO 
10 

9 

10 

I 

7 
6 

30   5.0 
40  6.6 
50   8.3 

4-? 
6-3 
7-9 

55 
56 

11 

59 

9-78853 
9.78  869 
9.78885 
9.78902 
9.78918 

IO 
I§ 

16 

16 
16 

9.89156 
9.89  177 
9.89  203 
9.89  229 
9.89255 

20 

26 

26 
26 
26 

0.10849 
0.10823 
0.10797 
0.10771 
0.10745 

9.89702 
9.89692 
9.89683 

9.89673 
9.89663 

IO 
10 

9 

10 
10 

5 
4 
3 

2 
I 

CO 

9-78934 

16 

9.89281 

26 

0.10719 

9  89653 

IO 

0 

Los*.  Cos. 

d. 

Loe.  Cot. 

c.  d. 

Log.  Tan. 

Loe.  Sin. 

d. 

f 

P.  P. 

r>2° 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

88° 


/ 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p.  p. 

0 

I 

2 

3 

4 

9.78  956 
9.78965 
9.78982 

9-78999 

16 
16 
16 

9.89281 
9.89307 

9.89359 
9.89385 

26 
26 
26 
26 

o.io  719 
0.10693 
0.10667 
0.10641 
o.io  615 

9.89653 
9.89643 

9.89633 
9.89623 
9.89  613 

9 
IO 
10 
10 

00 

57 
56 

l 

6 

8 
9 

9.79015 
9.79031 
9.79047 
9.79063 
9.79079 

16 
16 
16 

9.89411 
9.89437 
9.89463 
9.89489 
9.89515 

26 
26 
26 
26 

o.io  589 
o.  10  563 

0.10537 

o.io  511 

0.10485 

9.89604 
9.89594 
9-89584 

9.89  574 
9.89564 

10 
10 
10 
10 

55 
54 
53 
52 

26 

6     •?  6 

*\ 

0       P 

10 

II 

12 

13 

9.79095 

9-79  ni 
9.7912? 

9-79  143 
9-79  159 

16 
16 
16 
16 

9.89  541 
9.89  567 

9.89  593 
9.89619 
9.89645 

26 
26 
26 
26 

oA 

0.10459 

0.10433 

o.  10  407 
0.10381 

0.10355 

9.89554 
9.89544 

9-89534 
9.89  524 
9.89514 

9 

10 
10 
10 

50 

49 
48 

47 
46 

7     3-o 

8     3-4 

9     3-9 
10    4.3 
20     8.5 

2.5 
3-0 
3-4 
3-8 

8^5 

\l 

17 

18 

19 

9.79175 
9.79191 
9-7920? 
9.79223 
9.79239 

16 
16 
16 
16 

,£. 

9.89671 
9.89697 

9.89723 
9.89749 

9.89775 

26 
26 
26 
26 

0.10329 
o.  10  303 
0.10277 
0.10251 
o.io  225 

9.89504 
9.89494 
9.89484 

9.89474 
9.89464 

IO 
10 
10 
10 
10 

45 
44 
43 
42 

30  13.0 
40  17.3 
50  21.5 

12.? 

17.0 

21.2 

20 

21 
22 
23 
24 

9.79255 
9.79271 
9.7928? 
9.79303 
979319 

16 
16 
16 
16 

,£. 

9.89  801 
9.89827 

9.89853 
9.89879 
9.89905 

26 
26 

26 
26 

o.io  199 
o.io  173 
o.  10  147 

O.IO  121 

0.10095 

9.89454 
9.89444 
9.89434 
9.89424 
9.89414: 

IO 

10 
10 
10 
10 

40 

39 
38 

37 
36 

3 

27 
28 
29 

9-79335 
9-79351 
9.7936? 

9-79383 
9-79399 

16 
16 

16 

9.89931 
9.89957 
9.89982 
9.90003 
9.90034 

26 

25 
26 
26 

0.10069 
0.10043 

O.IO  01? 

0.09  991 
0.09  965 

9.89404 
9.89394 
9.89384 

9-89374 
9.89364 

IO 
10 
10 
10 
10 

35 

34 
33 
32 

Ig      I 

6     1-6     i 
7     1.9     i 

8       2.2       2 

6      iS 
.6      1.5 
•  8      1.8 

.1        2.6 

30 

32 
33 

34 

9.79415 
9-79431 
9-79446 
9.79462 

9-79478 

16 
15 
16 
16 

9.90066 
9.90086 

9.90  112 

9-90138 
9.90  164 

26 

26 
26 

25 

0.09  939 

0.09913 

0.09  88? 
0.09  861 
0.09  836 

9.89354 
9.89344 

9-89334 
9.89324 
9.89314 

IO 
10 
10 
IO 

16 

30 

29 
28 

27 
26 

9     2.5     2 

10       2.?       2 

20     5-5     5 
30     8.2     8 
40   n.o  10 

-4     2.3 
-6     2.6 

•3     5-i 
.0     7.? 
.6    10.3 

UINO  t^CO  ON 

CO  CO  CO  CO  CO 

9-79494 
9-79510 
9-79526 

9-79541 
9-79557. 

16 
16 

ii 

9.90  190 

9.90  216 

9.90242 

9  90  268 
9.90  294 

26 
26 
26 

26 

oP 

0.09  8  10 
0.09  784 
0.09  758 
0.09  732 
0.09  706 

9.89304 
9.89294 
9.89284 
9.89274 
9.89  264 

10 
IO 
10 
10 
rr» 

25 

24 
23 

22 
21 

12.9 

i 

42 

43 

44 

9-79573 
9.79589 
9.79605 
9.79626 
9-79636 

II 

\l 

9.90319 

9-90  345 
9.9037? 
9.9039? 
9-90423 

25 
26 
26 
26 
25 

0.09  686 
0.09  654 
0.09  623 
0.09602 
0.09  577 

9-89253 
9.89243 

9-89233 
9.89223 
9.89213 

10 
10 
IO 
10 

20 

19 

17 
16 

id    i 

6     .6 

; 

3    9,    ' 

o  0.9 

46 
47 
48 

49 

9-79652 
9.79668 
9-79683 
9.79699 
9.79715 

16 
16 

9.90449 

9.90475 
9.90  501 

9.90526 
9.90552 

26 
26 

3 

_/? 

0.09  551 
0.09  525 
0.09  499 
0.09473 
0.09  44? 

9.89  203 
9.89  193 
9.89  182 
9.89172 
9.89  162 

10 

16 

10 
10 

15 
H 
13 

12 
II 

7     .2 
8     .4 
9     -6     . 

10        .? 

20   3-5    3- 

T    i.i 
3   1.2 

5   '1 
6   '•$ 

3  3-i 

50 

52 
53 

54 

9.79736 

9-79748 
9-79762 

9-7977? 
9-79793 

'3 

16 
15 
15 
16 

9-90578 
9.  90  604 
9.90630 
9.90656 
9.90  682 

20 
26 

3 

26 

0.09  421 
0.09  395 
0.09  370 
0.09  344 
0.09  318 

9.89152 
9.89  142 
9.89132 

9.89  121 

9  89  1  1  1 

IO 
10 
10 

16 

10 

10 

9 
8 

1 

30   5-2    5. 
40   7.0   6. 
50   8.?    8. 

o  4.? 
6  6.3      i 
3  7.9 

1 

59 

9.79809 
9.79824 
9-79840 
9-79856 
9.79871 

16 

9.9070? 

9.90733 
9.90759 
9.90785 
9.90811 

25 
26 
26 
26 
25 

0.09  292 
0.09  265 
0.09  246 
0.09  214 
0.09  189 

9.89  101 
9.89091 
9.89081 
9.  89  076 
9.89066 

IO 

16 

10 

16 

10 

S 
4 
3 

2 
I 

60 

9-79887 

*S 

9.90837 

20 

0.09  163 

9.89056 

IO 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

.1. 

' 

P.  P. 

! 

51 


386 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

39° 


Log.  sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p.  ] 

P.                I 

0 

2 

3 
4 

9.79887 
9.79903 

9-79918 
9-79934 
9-79949 

16 

!l 

990837 
9.90863 
9.90883 
9.90914 
9.90940 

26 

25 
26, 

25 

0.09  163 
0-09137 
0.09  ii! 
0.09085 
0.09060 

9.89056 
9.  89  040 
9.  89  030 
9.89019 
9.89009 

10 
10 
10 
IO 

TO 

60 

59 
58 

i 

6 

8 
9 

9.79965 
9.79986 
9-79996 
9.8001! 
9.80027 

i§ 

9.90966 
9.90992 
9^101? 
9.91  043 
9.91  069 

26 

11 

26 

0.09034 
0.09008 
0.08  982 
0.08  956 
0.08  936 

9.88999 
9.88  989 

9.  88  978 
9.88968 
9.88958 

10 
10 
10 
10 

TO 

55 
54 
53 
52 

26 

6t  t 

'       „   a 

10 

ii 

12 

13 

9.  80  042 
9.80058 
9.80073 
9.80089 
9.80  104 

15 
15 

9.91  095 

9.91  121 

9.91  146 
9.91  172 
9.91  198 

26 
25 

0.08  905 
0.08  879 
0.08  853 
0.08  82? 
0.08  802 

9-8894? 
9.8893? 
9.88927 
9.88917 
9.88906 

10 
10 
10 
10 
in 

50 

49 
48 
47 
46 

2.C 

7     3-* 
8     3-< 
9    3-< 
10    4.: 

20      8  f 

)       2.5 

[     *4 
)     3-8 

1     4-2 

:       8  q 

15 

i6 

17 
18 

19 

9.  80  I  20 
9.80135 

9.  80  1  5  1 
9.80166 
9  80  182 

i$ 
if 

T? 

9.91  224 
9.91  250 
9.91  275 
9.91  30! 
9.91  327 

26 
25 

0.08  776 
0.08  750 
0.08  724 
0.08  698 
0.08  673 

9.88896 
9.88886 
9.88875 
9.88865 
9.88855 

10 
10 
10 
10 
TO 

45 
44 
43 
42 

30  i3-c 
40  17.' 

50   2I.( 

>       0.5 
)    12.? 

i  17-0 

5    21.2 

20 

21 

22 
23 
24 

9.80  19? 
9.80213 
9.80228 
9.80243 
9.80259 

1  y 

15 
l5 
1$ 

9-91  353 
9-91  378 
9.91  404 
9.91  430 
9.91  456 

25 
26 

2P 

0.08  647 
0.08  621 
0.08  595 
0.08  570 
0.08  544 

9.88844 
9.88834 
9.88823 
9.88813 
9.88803 

10 
10 
10 

10 

40 

39 
38 

37 
36 

25 
26 
27 
28 
29 

9.80274* 
9.80289 
9.80305 
9.80326 
9.80335 

15 

1$ 

15 

9.91  48! 
9.91  50? 

9-91  533 
9.91  559 
9.91  584 

5 
26 

25 
26 

25 

_/r 

0.08518 
0.08492 
0.08  467 
0.08441 
0.08415 

9.88792 
9.88782 
9.88772 
9.8876! 
9.88751 

IO 

16 

10 

16 

10 

35 
34 
33 
32 

16 

6     1.6 

7     1.8 
8     2.! 

*B     15 

2.6       2.0 

30 

32 
33 

1     34 

9.80351 
9-80366 
9.8038! 
9.80397 
9.80412 

15 
15 

9.91  616 
9.91  636 
9.91  662 
9.91  687 
9.91  713 

2O 

11 

25 

26 

-7? 

0.08  389 
0.08  364 
0.08  338 
0.08  312 
0.08  28g 

9.88746 
9.88730 
9.88720 
9.88709 
9.88699 

16 

10 

16 

10 

30 

29 
28 

27 
26 

9     2.4 

10       2.6 
20       5.3 

30     8.0 
40   10.6   i 

2-3       2.2 

2.6     2.5 
5-i     5-0 

7-?     7.5 
0.3    10.0 

3 

37 
38 

39 

980427 
9.80443 
9.80458 
9.80473 
9  80  48$ 

15 
15 

15 

T  p 

9-91  739 
9.91  765 
9.91  796 
9.91  8ig 
9.91  842 

25 
26 

2§ 

_  p 

0.08  261 
0.08  235 
0.08  209 
0.08  183 
0.08  158 

9.88683 
9.88678 
9.8866? 
9.88657 
9.88646 

16 
16 
16 
16 

25 

24 
23 

22 
21 

2.9    12.5 

40 

42 
43 
!     44 

9.80504; 
9.80519! 
9-8053$ 
9-80549 
9-80564 

J5 
15 
if 
15 
15 

9.91  867 
9.91  893 
9.91  919 
9.91  945 
9.91  976 

25 
26 

25 
26 

25 

0.08  132 
0.08  log 
0.08081 
0.08055 
o.  08  029 

9.88636 
9.88625 
9.88615 
9.88604 
9-88  594 

IO  . 
10 

16 

10 
10 

20 

19 
18 

17 
16 

ii 
6    i.i 

[5     10 

.6    i.o 

45 
46 
47 
48 
49 

9.80580 
9.80595 
9.80616 
9.80625 
9.  80  646 

i5 
i5 
15 
i5 

9.91  996 

9.92  022 
9.92  04? 
9.92073 
9.92099 

26 

11 

25 

0.08  004 
0.07  978 
0.07  952 
0.07  926 
0.07  901 

9-88  583 

9-88573 
9.88  562 

9-88  552 
9.8854! 

IO 

16 

10 

16 

10 

15 
H 
13 

12 
II 

lil 

9    i-6 
10    1.8 
20   3-6  3 

.2     I.! 

.4    1.3 
.6    1.5 

•?    1-6 
-5    3-3 

50 

52 
53 
54 

9.80655 
9.80671 
9.80686 
9.80701 
9-80716 

i5 
15 

9.92  12$ 
9-92  156 
9.92  176 
9.92  20! 
9.92  22? 

26 
25 

11 

0.07  875 
0.07  849 
0.07  824 
0.07  798 
0.07  772 

9-88  531 
9.88  526 
9.88510 
9.88499 
9.88489 

10 
10 

16 
16 

10 

10 

7 
6 

30   5.5   5 
40   7-3  7 
50   9.1   8 

.2    5.0 
•o   6.6 
.?   8.3 

1  1 

59 

9.8073? 
9-80746 
9.8076! 

9.80776 
9.80  79! 

15 
15 
15 
15 
15 

9.92253 

9.92  278 

9.92304 

9-92  33° 

25 
26 

25 
25 

x- 

0.07  747 
0.07  721 
0.07  695 
0.07  670 
0.07  644 

9-88478 
9.8846? 
9.88457 

9-88446 
9.88436 

10 

ii 

10 

16 
16 

5 
4 
3 

2 

GO 

9.80806 

15 

9.92  38! 

26 

0.07613 

9.88425 

10 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

' 

5O 


387 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

4O° 


Log.  Sin. 

d. 

Log.  Tnn. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p.  p. 

0 

I 

2 

3 

4 

9.80805 
9.80822 
9.80837 
9.80852 
9.80867 

15 
15 
15 

9.9238! 
9.92407 
9-92432 

9-92458 
9.92484 

25 
25 
26 

25 

0? 

0.07613 

0.07  593 
0.07  567 
0.07  541 
0.07  516 

9.88425 
9.88415 
9.88404 
9-88393 
9.88383 

16 
ii 

10 
10 

GO 

8 
9 

9.80882 
9.80897 
9.80912 
9.80927 
9.  80  942 

15 
15 
15 
15 

9.92  509 

9.92  535 
9.92  561 

9.92  586 
9.92612 

25 

3 

25 
25 

0.07  490 
0.07  465 
0.07  439 
0.07413 
0.07  388 

9.88372 
9.88361 
9.88351 
9.88346 
9.88329 

ii 

10 

16 

1  1 

55 
54 
53 
52 

26 

6             (• 

25 

_  o 

10 

ii 

12 

13 

14 

9.80957 
9.80972 
9.80987 

9.81  ooi 
9.81  015 

Z5 
15 
I| 

15 

9.92  638 
9.92  663 
9.92689 
9.92714 
9.92  746 

25 
25 

t 

~? 

0.07  362 
0.07  335 
0.07311 
0.07  285 
0.07  259 

9.88319 
9.88303 
9.8829? 
9.88287 
988275 

10 

1  1 

10 
10 

50 

49 
48 

47 
46 

2.D 

7     3-o 
8     3-4 
9     3-9 
io     4.3 
20     85 

2.5 
3-o 
3-4 

3-8 
4.2 
8  5 

17 

18 
19 

9.81  031 
9.81  045 
9.81  061 
9.81075 
9.81  091 

15 
15 
15 

9.92  766 
9.92791 
9.92817 
9.92  842 
9.92863 

25 
25 
25 

i 

09 

0.07  234 
0.07  203 
0.07  183 
0.07  15? 
0.07  131 

9.88265 
9.88255 
9.88  244 
9.88233 
9.88223 

10 
10 

II 

10 

45 
44 

43 
42 

30   13-0 
40   17.3 

50  21.6 

17.0 

21.2 

20 

21 
22 

23 
24 

9.81  106 

9.8l  121 

9.81  136 
9.81  156 
9.81  165 

15 
15 

15 

9.92894 
9.92919 
9.92945 
9.92971 
9.92  996 

25 

25 
25 
26 

25 
o? 

0.07  1  06 
0.07  086 
0.07055 
0.07  029 
0.07  003 

9.88212 
9.88201 
9.88  196 
9.88  1  80 
9.88  169 

10 

1  1 

10 

II 

Tri 

40 

I 

25 
26 

27 
28 
29 

9.81  1  86 
9.81  195 

9.8l  210 
9.8l  225 

9-  8  1  239 

14 
15 
15 

9.93022 
9-9304? 
9-93073 
9-93098 
9.93124 

25 

25 

25 
25 
26 

~? 

0.06978 
0.06  952 
0.06927 
0.06  901 
0.06875 

9.88153 
9.88  147 

9-88  137 
9.88  126 
9.88115 

II 
10 
II 
10 

35 

34 
33 
32 

X5     s< 

6     1.5      i. 
7      1.8      i. 

8       2.6       2. 

5      1-4 
?     i.7 
o      1.9 

30 

32 
33 
34 

9.81  254 
9.81  269 
9.81  284 
9.81  299 
981  313 

15 
15 

9.93150 

9-93  175 
9.93201 
9.93226 
9.93  252 

25 
25 
25 
25 
25 

0.06  850 
0.06  824 
0.06  799 
0.06  773 
0.06  748 

9.88  1  04 
9.  88  094 
9.88083 
9.88072 
9.88061 

16 
ii 
ii 
16 

30 

29 
28 

27 
26 

9     2-3     2- 

10       2.6       2. 
20        5.T        5. 
30       7-?        7- 

40   10.3    io. 

2       2.2 

5     2-4 
o     4.8 
5     7.2 
o     9-6 

5T  0      T 

i-nNO  tv.OO  ON 
CO  CO  CO  CO  CO 

9.81  323 
9-  8  1  343 
9.81  358 
9.81  372 
9  8  1  387 

ii 

a 

9-93  278 
9-93  303 
9-93329 
9.93  354 
9-9338o 

25 
25 
25 
25 
o? 

0.06  722 
0.06695 
0.06671 
0.06  645 
0.06620 

9.88056 
9.88039 
9.88029 
9.88018 
9.88007 

ii 
16 
ii 
ii 

25 

24 
23 

22 
21 

12.1 

40 

42 
43 
44 

9.  8  1  402 
9.81  416 
9.81  431 
9.81  446 
9.81  466 

15 

a 

14 

9-93405 
9-93431 
9-93456 
9.93482 

9-93  5°8 

25 
25 
25 

0? 

0.06  594 
0.06  569 
0.06  543 
0.06  518 
0.06  492 

9.87996 
9.87985 
9.87974 
9.87963 
9.87953 

ii 
ii 
ii 
16 

20 

I  Q 

17 
16 

II 

6     .1 

10 

i.o 

46 
47 
48 

49 

9.81475 
9.81  490 
9.81  504 
9.81519 
9.81  534 

'* 

i] 

9-93  533 
9-93  559 
9-93  584 
9.93610 

25 
25 
25 

25 
25 

0.06  466 
0.06  441 
0.06415 
0.06  390 
0.06  364 

9.87942 
9.87931 
9.  87  920 

9.87909 
9.87893 

1  1 
ii 
ii 
16 
ii 

15 
.14 

13 

12 
II 

7     -3 
8    .4 

9     -6 
io     .3 

2o   3-6 

1.2 
1.4 

1.6 
i.? 
3-5 

50 

52 
53 

54 

9-  8  1  548 
9.81  563 
9.81  578 
9.81  592 
9.81  607 

a 

9.93661 
9.93686 
9.93712 

9-9373? 
9.93  763 

25 
25 
25 

0.06  339 
0.06313 
0.06  288 
0.06  262 
0.06  237 

9.8788? 

9.87875 
9.87865 
9.87854 
9.87844 

1  1 
ii 
ii 
ii 

10 

10 

7 
6 

30   5-5 
40   7-3 
50   9.1 

5.2 
7.0 

8.? 

59 

9.81  62! 
9.81  636 
9.81  656 
9.81  665 
9.81  680 

a 

a 
15 

9.93814 
9.93840 

9-93  865 
9.93  891 

25 

11 

25 
25 

0.06  211 

0.06  1  86 
0.06  160 
0.06  134 
0.06  109 

9.87833 
9.87822 
9.87811 
9.87800 
9.87789 

ii 
ii 
ii 
II. 
ii 

5 
4 
3 

2 

60 

9.81  694 

H 

9-93  916 

25 

0.06  083 

9-87778 

ii 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tnn. 

Log.  Sin. 

d. 

' 

P.  P. 

49 


388 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

41° 


, 

Log.  Sin.       d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos.        d. 

p.  p.              1 

0 

I 

2 

3 
4 

9.8l  694 
9.81  709 
9.81  723 
9.8l  738 
9.8l  752 

4 
H 

14 

4 
14 

14 
4 
4 
4 
14 
14 

4 
4 

4 

4 
4 

4 
4 

9939*6 
9-93942 

9-93993 
9-94018 

25 
25 
25 
25 

25 
25 
25 

25 
25 

25 
25 
25 
25 
25 
25 
25 
25 

25 
25 
25 
25 

25 
25 
25 
25 
25 
25 

25 
25 
25 
25 
25 

25 

25 
25 

25 
25 
25 

2? 
25 

11 

25 
25 
25 

25 
25 

0.06083 
0.06058 
O.o6  032 
6.06007 
0.05  981 

9.87778 
9.87767 
9.87756 

9.87  745 
9.87734 

II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 
II 

00 

59 
58 

57 
56 

6 

8 
9 

10 
20 
30 
40 
50 

6 

8 
9 

10 

20 
30 
40 
So 

6 

8 
9 

10 
20 
30 
40 
50 

(ui 

\8 

25 

2-5 
3-o 
3-4 
3-8 
4-2 
8-5 

12.? 

17.0 

21.2 

M 

1.4 
1.7 

*  9 

2.2 
2.4 

4-8 
7.2 

9-6 

12.  1 

II 

•  7 

•9 
3-8 
5-? 
7-6 
9.6 

^ 

OF 
*IV 

^A! 

25 

2.5 
2.9 

3-3 

&3 

12.5 

20.3 

14 

1.4 

1-6 
1-8 

2.1 
2.3 

4-6 

7.0 

9-3 
1  1.6 

ii 
i.i 

a 
5.5 

7-3 
9.1 

THK            r 

ERSIT1 

8 
9 

9.81  767 
9.81  781 
9.81  796 

9.81  816 
9.81824 

9.94044 

9-94.P69 
9.94095 
9.  94  1  26 
9.94  146 

9-94I7I 
9-94  197 
9.94222 
9.94248 
9-94273 

0.05  956 
0.05  936 
0.05  905 
0.05  879 
0.05  854 

9.87723 
9.87712 
9.87  701 
9.87690 
9.87679 

55 
54 
53 

52 

10 

ii 

12 
13 

!     14 

9.81  839 
9-8i  853 
9.81  868 
9.81  882 
9.81  897 

0.05  823 
0.05  803 
0.05  77? 
0.05752 
0.05  725 

9.87668 
9.87657 

9.87645 
9.87634 
9.87623 

50 

49 
48 
47 
46 

J! 

17 
18 

19 

9.81  911 
9.81  925 
9.81  940 
9.81  954 
981  o6g 

9.94299 
9.94324 
9.94350 

9-94375 
9.94406 

0.05  701 
0.05  675 
0.05  650 
0.05  625 
0.05  599 

9.87612 
9.87601 

9-87  590 
9.87579 
9.87  568 

45 
44 
43 

42 

20 

21 
22 
23 
24 

9-  8  1  983 
9.8199? 
9.82012 
9.8202^ 
9.  82  046 

9.94426 
9.94451 
9-94477 
9-94  502 
9.94528 

0.05  574 
0.05  548 
0.05  523 
0.05  49? 
0.05  472 

9-87  557 
9-87  546 
9-87  535 
9.87523 

9-875*2 

40 

* 

I 

25 
26 

27 
28 

29 

9.82055 
9.82069 
9.82083 
9.82098 
9  82  112  i 

9-94553 
9-94579 
9.94604 
9.94630 
9.94655 

0.05446 
0.05  421 
0.05  395 
0.05  370 
0.05  344 

9-87  50* 
9.87496 
9.87479 
9.87468 
9-87457 

35 
34 
33 
32 

30 

32 
33 
34 

982125, 
9.82  140  i 
982155 
9.82  169 
9.82  183 

9.94681 
9-94706 
9-94732 
9-94757 
9-94782 

0.05  3r9 
0.05  293 
0.05  268 
0.05  243 
0.05  217 

9.87445 

9.87434 
9.87423 
9.87412 
9.87401 

30 

29 
28 

27 
26 

35 
36 
37 
38 
39 

9.82  19? 

9.82  212 
9.82  226 
9.82  246 
9-82  2:54: 

H 
4 
14 
4 
*4 
4 
14 
14 

H 
14 

4 

14 
14 
14 
14 
4 
H 

14 

9.94808 

9-94833 
9.94859 
9.94884 
9.94910 

0.05  192 
0.05  165 
0.05  141 
0.05  n§ 

0.05  090 

9.87  389 
9-87  378 
9.87  367 

9.87  356 
9-87345 

25 

24 
23 

22 
21 

40 

42 
43 
44 

9.82  269  i 
9.82  283 
9.82297 
9.8231! 

9-94935 
9.94961 
9.94986 
9.95011 
9.95037 

0.05  064 
0.05  039 
0.05  014 
0.04  983 
0.04  963 

9-87333 
9-87322 
9.87311 

9-87  3°° 
9-87288 

20 

*9 
18 

17 

16 

45 
46 

47 
48 
49 

9-82354 
9.82368 
9.82382 
9.82396 

9.95062 
9.95088 
9-95II3 
9-95  139 
9-95  164 

0.04  93? 
0.04  912 
0.04  885 
0.04  86  1 
0.04  836 

9.87  277 
9.87  266 

9-87  254 
9.87243 
9.87232 

*5 
*3 

12 

II 

50 

52 
53 
54 

9.82416 
9.82424 

9-82438 
9.82452 
9.82467 

9-95  189 
9.95215 
9.95  246 
9.95  266 
9-95  291 

0.04  816 
0.04  785 
0.04759 
0.04734 
0.04  703 

9.87  221 

9.87  198 
9.87  187 
9.87175 

10 

I 

7 
6 

55 
56 

59 

9.82481 
9.82495 
9.82  509 
9.82  523 

9-82  537 

9-953*6 
9-95  342 
9-95  36? 
9-95  393 
9-954*8 

0.04  683 
0.04658 
0.04632 
0.04  607 
0.04  581 

9.87  164 

9-87*53 
9.87  141 
9.87  130 

9.87  H8 

5 
4 
3 

2 

I 

00 

9.82551 

9-95443 

0.04  556 

9.87  10? 

0 

Log.  Cos.        d. 

Log.  Cot.     c.  d.     Log.  Tan. 

Log.  Sin. 

d. 

p.  P. 

48' 


389 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

42° 


Log.  Siii. 

d 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p.  p 

0 

I 

2 

3 

4 

9.82551 
9.82565 

9.82579 
9.82593 
9.82607 

14 
14 

9-95443 
9.95469 

9-95  494 
9-95  520 
9-95  545 

25 

25 
25 

0.04  555 
0.04  531 
0.04  505 
0.04480 
0.04454 

9.87  io? 

9.87  096 
9.87084 
9.87073 
9.87062 

II 
II 
II 
II 

00 

p 
i 

6 

8 
9 

9.82621 
9-82635 
9.82  649 
9.82663 
9.82677 

H 
H 
H 
H 

9-95  57i 
9-95  596 
9.95621 
9.95647 
9.95672 

25 

25 
25 

25 

0.04429 
0.04404 
0.04  378 
0.04353 
0.04  32? 

9.87056 

9.87039 
9.8702? 

9.87016 
9.87  oo4 

II 
II 
II 
II 

55 

54 

53 
52 

6_    p 

25 

10 

ii 

12 
13 
14 

9.82691 
9.82705 
9.82719 

9.82745 

H 

i3 

9-9569? 
•9-95  723 
9-95  748 
9-95774 
9-95  799 

25 

25 
25 
25 
25 

0.04302 
0.04  277 
0.04  251 
0.04  226 

0.04  200 

9.86993 

9.86982 
9.86976 
9.86959 
9.8694? 

II 
II 
II 
II 

T? 

50 

49 
48 
47 
46 

2.5 
7     3-o 
8     3-4 
9     3-8 
10     4.2 
20     85 

2.5 
2,9 

1! 

8  3 

17 

18 
19 

9.82  766 
9.82774 
9-82788 
9.82802 
9.82816 

H 

9.95  824 
9.95850 

9.95901 
9-95  926 

25 

25 
25 
25 
25 

0.04  175 
0.04  150 
0.04  124 
0.04  099 
0.04074 

9.86936 
9.86924 
9.86913 
9.86901 
9.86890 

II 
II 
II 
II 

45 
44 
43 
42 

30   12.7 
40   17.0 

50    21.2 

12.5 
16.5 

20.§ 

20 

21 

22 
23 
24 

9.82830 
9.82  844 
9.82858 
9.8287? 
9.82885 

H 
H 
13 
14 

9.9595! 

9-95977 
9.  96  002 
9.9602? 
9-96053 

25 
25 
25 

0? 

0.04043 
0.04023 

0.03  99? 
0.03  972 
0.03  947 

9-86878 
9.86867 
9.86855 
9.86844 
9.86832 

II 
II 
II 
II 
j  « 

40 

39 
38 

i 

25 
26 

27 
28 
29 

9.82899 
9.82913 
9.82927 
9.82  946 
9-82954 

13 

H 

9.96073 
9.96  104 
9.96  129 
9.96154 
9.96  1  80 

25 

25 
25 
25 
25 

7  C 

0.03  921 
0.03  896 
0.03  871 
0.03  845 
0.03  820 

9.86821 
9.86809 
9.86798 
9.86786 
9.86774 

II 
II 

12 
II 

35 
34 
33 
32 

6     1.4 
7     1-6 
8     i-8 

13 

1.6 
1.8 

30 

32 
33 
34 

9.82963 
9.82982 
9.82996 
9.83009 
9.83023 

13 
H 
13 
14 

9.96  205 
9.96  236 
9.96256 
9.96281 
9.96305 

25 
25 

25 
25 

25 
i? 

0.03  795 
0.03  769 
0.03  744 
0.03713 
0.03  693 

9-86763 
9.86751 
9.  86  740 
9.86723 
9-86715 

II 
II 
II 
12 

T  =• 

30 

29 
28 

27 
26 

9    2.1 
10     2.3 

20      4.5 
30      7.0 

40     9-3 

2.0 

2.2 

4-5 
6.? 
9.0 

35 
36 

s 

39 

9.83037 
9.83051 
9.83064 

9-83078 
9.83092 

i3 
H 
13 

9.96332 
9-9635? 
9-96383 
9.96408 

9-96433 

25 

25 
25 

2P 

0.03668 
0.03  642 
0.03617 
0.03  592 
0.03  565 

9.86  705 
9.86693 
9.86682 
9.86676 
9-86658 

1  1 

II 
II 
II 
12 

25 
24 
23 

22 
21 

50  11.5 

42 
43 
44 

9.83  1  06 
9.83119 

9-83I33 
9.83  147 

9.83  1  66 

14 
13 

9-96459 
9.96484 

9-96  5°9 
9-96535 
9.96560 

5 
25 
25 
25 
25 

0? 

0.03  541 
0.03  516 
0.03  496 
0.03465 
0.03  440 

9.86647 
9.86635 
9.86623 
9.86612 
9.  86  606 

1  1 
II 
12 
II 
II 

20 

19 
.  18 
17 
16 

12 
6       1.2 

if     ii 

[.i    i.i 

49 

9-83  174 
9.83  1  88 
9.83  201 
9.83215 
9.83229 

iS 
iS 

9-96  585 
9.96611 
9.96636 
9.96661 
9.96687 

25 
25 
25 

25 

0.03414 

°.°3  389 
0.03  364 
0.03  338 
0.03313 

9-86  588 
9.86  577 
9.86  565 

9-86553 
9.86542 

12 
II 

II 
12 
II 

15 
13 

12 
II 

7     1-4 
8      1.6 
9     1.8 

IO       2.0 

20     4.0  ; 

i.3  1.3  ' 

[-5    1-4 
1.7    1.6 
[.9    1.8 
5-8   3-6 

50 

52 
53 

54 

9.83242 
9.83256 
9.83  269 
9-83283 
9.83297 

13 
'3 

9.96712 
9.9673? 
9-96763 
9-96788 
9.96813 

25 
25 
25 
25 
25 

0.03  28? 
0.03  262 
0.03  237 
0.03  21  T 
0.03  185 

9.86  530 
9-86518 
9.86  507 
9.86495 
9.86483 

12 
II 
II 
12 
II 

10 

9 
8 

6 

30     6.0 
40     8.0  \ 
50   10.0  c 

>•?  5-5 

'•&   7-3 
).6   9-1 

1 

59 

9.83316 
9-83324 
9-8333? 
9-8335I 
9.83365 

13 
13 
14 

9-96839 
9.96  864 
9.96889 
9.96915 
9.96  946 

25 
25 
25 

0.03  161 
0.03  135 
0.03  1  16 
0.03085 
0.03  059 

9.86471 
9.86460 
9.86448 
9.86435 
9.86424 

12 
II 
12 
II 
12 

5 
4 
3 

2 

j    60 

9-83378 

[3 

9.96965 

*5 

0.03  034 

9.86412 

12 

~0~ 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

f 

P.  P 

j 

TABLE  VII.—  LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

43° 


/ 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

p.  p. 

0 
I 

2 

3 
4 

9-83378 

9-83405 
9.83419 

13 

9.96965 
9.96991 
9.97016 
9-9704T 
9.97067 

25 
25 

25 
25 

0.03034 
0.03009 
O.02  984 

0.02  958 
0.02  933 

9.86412 
9.86401 
9.86389 
9.8637? 
9.86365 

II 
12 
II 
12 

GO 

P 
g 

S 

9 

9.83446 
9-83459 
9-83473 
9.83485 
9.83500 

13 
13 
13 

9.97092 
9-97-ii? 
9-97  H3 
9.97  1  68 

9-97  193 

25 
25 
25 
25 

25 

~P 

O.O2  908 
0.02  882 
0.02  857 
0.02  832 
0.02  805 

9.86354 
9.  86  342 
9.86  330 
9.86313 
9.86305 

12 
12 
II 
12 

55 
54 
53 
52 

23 

62  £ 

25 

2r 

10 
ii 

12 

13 

14 

9-835I3 
9.83527 

9-83  540 
9.83554 
9.83  567 

13 
13 
13 

9-972I9 
9-97  244 
9.97269 
9.97295 
9.97320 

25 
25 
25 
25 

25 
^p 

0.02  781 
0.02  756 
0.02  736 
0.02  705 
0.02  680 

9.86294 
9.86  282 
9.86271 
9.86  259 
9.86  247 

12 
II 
12 
12 

n" 

50 

49 
48 
47 
46 

7     3-o 
8     3-4 
9     3-8 
10     4.2 

20       8.5 

•5 

2.9 

ii 

4.i 
8.3 

15 

16 

17 
18 

19 

9.83  586 

9-83  594 
9.8360? 
9.83621 
9-83634 

13 
13 
13 
13 

9-97  345 
9-97  370 
9-97  396 
9.97421 

9-97446 

25 

3 

25 
25 

0.02  654 
0.02  62§ 
O.02  604 

0.02  578 
0.02  553 

9-86235 
9.86223 
9.86211 
9.86  199 
9.86  187 

12 
12 
12 
12 

T? 

45 
44 

43 

42 

30    12.? 
40    17.0 
50    21.2 

12.5 

16.6 
20.3 

20 

21 

22 

23 
24 

9.83647 
9.83661 
9.83674 
9.83688 
9.83701 

I3 
13 

13 

9-97472 
9-9749? 
9-97  522 
9-97  548 
9-97  573 

25 

25 
25 

25 

0.02  528 
0.02  502 

0.02  477 

0.02  452 
0.02  427 

9.86  176 
9.86164 
9.86  152 
9.86  140 
9.86  128 

12 
12 
12 
12 
1  2 

40 

39 
38 

25 
26 

27 
28 

29 

9.83714 
9.83728 
9.83741 

9.83754 
9.83768 

I3 
13 
13 
13 

13 

9.97  598 
9.97624 

9-97649 
9-97  674 
9.97  699 

25 
25 

25 
25 

2? 

0.02  401 
0.02  376 
0.02  351 
0.02  325 
0.02  306 

9.86  116 
9.86  104 
9.86092 
9.86080 
9.86068 

12 
12 
12 
12 

35 
34 
33 
32 

6     1.3 
7     1.6 
8     1.8 

13 

1.5 
i.f 

30 

32 

33 

!     34 

9.83781 

9.83794 
9.83808 
9.83821 
9  83  834 

13 
13 
13 

9.97725 
9.97  750 
9-97775 
9.97801 
9.97  826 

*y 

25 
25 

25 

25 

«p 

0.02  275 
O.O2  249 
0.02  224 
0.02  199 
0.02  174 

9.86056 

9.86044 
9.86032 
9.86020 
9.86008 

12 
12 
12 
12 
12 

30 

29 

28 

27 
26 

9     2.0 

10      2.2 
20      4.5 

30    6.? 

4O      9.0 
CO     T  T    5 

1.9 

2.f 
6J 
IO  Q 

35 
36 

H 

39 

9.83847 
9.83861 
9.83874 
9.83887 
9.83906 

13 
13 

9.97851 
9.97877 
9.97902 
9-9792? 
9.97952 

25 
25 
25 
25 
25 
<•»? 

0.02  148 

0.02  123 
O.02  098 

0.02  04? 

9.85  996 
9.85984 
9.85972 
9.85  960 
9.85948 

12 
12 
12 
12 

25 
24 
23 

22 
21 

40 

42 
43 
44 

9.83914 

9-83927 
9.83946 

9-83953 
9-83967 

13 
13 

13 

9.97978 
9.98003 
9.98023 
9.98054 
9.98079 

25 
25 
25 

25 
25 

0? 

0.02  022 

o.oi  995 
o.oi  971 
o.oi  946 
o.oi  921 

9.85936 
9.85924 
9.85912 
9.85900 
9.85  887 

12 
12 
12 
12 

20 

19 
18 

17 
16 

12         ! 
6       1.2 

[2      II 

1.2     I.I 

45 
46 

47 
48 

49 

9.83980 

9-83993 
9.84005 
9.84019 
9-84033 

13 
13 
13 

9.98  104 
9.98129 
9.98155 
9.98  1  86 
9.98205 

25 
25 
25 
25 
25 

o.oi  895 
o.oi  876 
o.oi  845 
o.oi  819 
o.oi  794 

9-85  863 
9.85851 

9-85  839 
9.85  827 

12 
12 
12 
12 

15 
13 

12 
II 

7      i-4 
8      1.5 
9     i-9 

IO       2.1 

20     4.  T 

1.4    i-3 
1.6   1.5 
1.8    1.7 
2.0   1.9 
*-o  3-8 

50 

52 
53 
54 

9.  84  046 
9.84059 
9.84072 
9.84085 
9.84093 

I3 
13 

13 
13 

9.98231 
9.98256 
9.98  281 
9.98  305 
9-98  332 

25 

25 

25 
25 
25 

o.oi  769 
o.oi  744 

O.OI  7I§ 

o.oi  693 
o.oi  668 

9.85815 
9-85  803 
9.85791 
9-85778 
9.85765 

12 
12 
12 
12 

10 

9 
8 

6 

30     6.2     i 
40     8.3 
50   10.4   u 

l:°  1:1 

D.O    9.6 

P 
t 

59 

9.84111 
9.84  124 
9.84138 
9.84151 
9.84  164 

13 

13 
13 

9-98  35? 
9.98382 
9.98408 
9-98433 
9-98458 

i 
n 

o.oi  642 
o.oi  61? 
o.oi  592 
o.oi  567 
o.oi  541 

9-85754 
9.85742 
9.85730 
9.85718 
9.85705 

12 

12 
12 
12 
12 

5 
4 
3 

2 

60 

9.84  177 

l* 

9.98483 

2  5 

o.oi  515 

9.85693 

12 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Siu. 

d. 

' 

p.  P 

391 


TABLE  VII.— LOGARITHMIC  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

44° 


/ 

Log.  Sin. 

d. 

Log.  Tan. 

c.  d. 

Log.  Cot. 

Log.  Cos. 

d. 

P.  P. 

0 

I 

2 

3 
4 

9.84177 
9.84  190 
9.84203 

9.84216 
9.84229 

13 

1.3 

13 
13 

9.98483 
9.98  509 
9-98  534 
9.98  559 
9.98  585 

25 
25 
25 
25 

2  C 

0.01  515 
o.oi  491 
o.oi  465 
o.oi  446 
o.oi  415 

9-85693 
9.85681 
9.85669 
9.85657 
9.85644 

12 
12 
12 
12 

GO 

59 

58 
57 
56 

1 

8 
9 

9.84242 
9.84255 
9.84263 
9.8428! 
9.8429^ 

J3 
13 
13 
13 
13 

9.98610 

9-98635 
9.98666 
9.98686 
9.98711 

Z5 
25 
25 
25 
25 
•?£ 

o.oi  390 
o.oi  364 
o.oi  339 
o.oi  314 
o.oi  289 

9.85632 

9.85  620 

9.85608 

9-85  595 

9-85  583 

12 
12 

12 
12 

TO 

55 
54 
53 
52 
5i 

2S 

_    p 

2S 

10 

ii 

12 

13 

H 

9-8430? 
9.84328 

9-84333 
9.84346 
9-84359 

I3 
13 
13 
13 
13 

9-98736 
9.98  762 
9.98787 
9.98812 
9-9883? 

25 
25 
25 
25 

25 
22 

o.oi  263 
o.oi  238 
o.oi  213 
o.oi  18? 
o.oi  162 

9-85  571 

9.85  559 
9-85  546 
9-85  534 
9.85  522 

12 

12 
12 
12 

50 

49 
48 

47 
46 

8 
9 

10 

20 

2.5 

3-o 

$ 

4.2 
8  c 

2.5 

3 

3.? 
4-t 

8  5 

15 

16 

17 
18 

19 

9-84372 
9-84385 

9-84398 
9.84411 
9.84424: 

J3 
13 
13 
13 

13 
Ta 

9.98  863 
9.98888 
9.98913 
9-98938 
9-98964 

*J 

25 
25 

25 

25 

•?B 

o.oi  137 

O.OI   112 

o.oi  085 
o.oi  06! 
o.oi  036 

9.85  509 

9-8549? 
9.85485 
9.85472 
9.85466 

12 
12 
12 
12 

45 

44 
43 
42 

4i 

30 
40 

50 

12.? 
17.0 
21.2 

"O 

12.5 
i6.g 

20.  g 

20 

21 

22 

23 

24 

9-84437 
9.84450 
9.84463 
9.84476 
9-84489 

13 

13 
13 
13 
I  ^ 

9.98989 
9.99014 
9.99040 
9.99065 
9.99096 

25 

25 

25 

25 
25 

21 

O.OI  OIO 

o.oo  985 
o.oo  960 
o.oo  935 
o.oo  909 

9.85448 
9-85435 
9-85423 
9.85411 

9-85  398 

12 
12 
12 
12 
12 

40 

38 

1 

25 
26 

27 
28 

29 

9.84502 
9.84514 
9-8452? 
9.84546 

9-84553 

1  j 
12 

13 
13 
13 
i3 

9.99115 
9.99  141 
9.99  1  66 

9-99  191 
9.99215 

*3 

25 
25 
25 

25 
2? 

o.oo  884 
o.oo  859 
o.oo  834 
o.oo  8o§ 
o.oo  783 

9-85  386 
9-85374 
9.85361 

9-85  349 
9-85  336 

12 
12 
12 
12 

T3 

35 
34 
33 
32 
3i 

[.« 

\l 

13 

1-3 

1.6 

1.8 

13 

1-3 
1.5 
1.7 

30 

3i 
32 
33 

!    34 

9.84566 
9.84579 
9.84592 
9.84604 
9.8461? 

13 
13 
12 

13 

9.99242 
9-99267 
9-99292 
9.99318 

9-99343 

25 

25 
25 

2i 

25 

~? 

0.00758 
o.oo  733 
o.oo  70? 
o.oo  682 
0.00657 

9-85  324 
9.85312 
9.85  299 
9.85  287 
9.85  274 

12 
12 
12 
12 

30 

29 

28 

27 
26 

1  9 
10 

20 

30 
40 

2.0 
2.2 
4-5 

6.? 
9.0 

Uft 

I.§ 
2.1 

4-3 

Ij 

35 
36 

% 

39 

9.  84  636 
9.84643 
9.84656 
9.84669 
9.84681 

I3 

12 
13 
13 
12 

9-99368 
9-99  393 
9.99419 
9.99444 
9.99469 

25 
25 
25 
25 
25 

o.oo  631 
0.00605 
o.oo  581 
o.oo  556 
o.oo  536 

9.85  262 
9.85  249 
9.85  237 
9.85  224 

9.85  212 

12 
12 
12 
12 
T» 

25 
24 

23 

22 
21 

5° 

.2 

lo.g 

40 

4i 

42 
43 
44 

9.84694 
9.84707 
9.84720 
9.84732 
9-84745 

X3 

12 

13 
12 

13 

Ta 

9.99494 
9-99  520 
9-99  545 
9.99576 

9-99  595 

25 
25 
25 
25 

25 
2p 

o.oo  505 
0.00480 

0.00455 

0.00429 
o.oo  404 

9-85  199 
9.85  187 
9.85  174 
9.85  l62 
9-85  149 

I§ 
12 
12 
12 
-a 

20 

19 

18 

17 
16 

g 

IS 

T    5 

12 

,T      ^ 

45 
46 

47 
48 

49 

9.84758 
9.84771 
9.84783 

9-8479S 
9.84809 

13 
12 
13 
12 

9.99621 
9.99645 
9.99671 
9.99697 
9-99722 

5 
25 
25 
25 

25 
_P 

o.oo  379 
0.00353 
o.oo  328 
o.oo  303 
o.oo  278 

9.85  137 
9.85  124 
9.85  112 
9.85099 
9.85087 

12 
I§ 

i2 

12 

12 

15 

H 
13 

12 
II 

I 

9 

10 

20 

1.4 
1-8 

1.9 

2.1 
4.1 

1.4 

1.6 

1.8 

2.0 
4.0 

50 

Si 
52 
53 
54 

9.84822 
9.84834 
9.84847 
9.84860 
9.84872 

J3 

12 
12 

13 
12 

¥  % 

9-9974? 
9.99772 

9^9798 
9.99823 
9.99843 

25 
25 
25 
25 

25 

o.oo  252 

O.OO  22? 
0.00  202 

o.oo  177 
o.oo  151 

9.85074 
9.85062 
9.85049 
9.85037 
9.85024 

12 
I§ 
12 
12 
13 

10 

6 

30 
40 

50 

6.2 

8.3 

10.4 

6.0 

8.0 

10.0 

P 
11 

59 

9.84885 
9.84898 
9.84916 
9.84923 
9.84936 

12 

J3 

12 
12 
13 

9-99873 
9.99899 
9.99924 
9-99949 
9-99974 

25 
25 
25 
25 

25 
_  P 

o.oo  125 

0.00  101 

0.00076 
0.00056 
0.00025 

9.85011 
9.84999 
9.84986 
9-84974 
9.84961 

12 
12 
12 
12 
13 

5 
4 
3 

2 

I 

60 

9-84948 

12 

0.00000 

25 

0.00000 

9-84948 

12 

0 

Log.  Cos. 

d. 

Log.  Cot. 

c.  d. 

Log.  Tan. 

Log.  Sin. 

d. 

/ 

p.  p. 

45 


392 


TABLE   VIII. 

LOGARITHMIC  VERSED  SINES  AND  EXTERNAL 

SECANTS. 


TABLE   VIII.— LOGARITHMIC   VERSED   SINES    AND    EXTERNAL   SECANTS. 

0°  1° 


/ 

Log.  Vers. 

& 

Log.  Exsec. 

D 

Log.  Vers. 

D 

Log.  Exsec. 

D 

' 

0 
I 

2 

3 

4 

CO 

2.62642 
3.22848 
3.58o6§ 
3-83054 

60206 

35218 
2498? 

—  CO 

2.62642 

3.22848 

3.58065 

3-83054 

60206 
35218 
2498? 

6.18271 
.19707 
.21119 
.22509 
.23877 

H35 

1412 

I38§ 
1368 

6.18278 
.19714 
.21126 
.22515 
.23884 

1436 
1412 
1390 
I368 

0 

2 

3 

4 

I 

8 
9 

4.02436 
.18272 
.31662 
.43260 
•  53490 

15836 
13389 
11598 
10236 

4.02435 
.18272 
.31662 
.43266 
.  53491 

19382 
15836 
I338§ 

1  1  598 
10236 

6.25223 
•26549 
.27856 
.29142 
.30416 

'346 
1326 

1305 

I28g 

1268 

6.25231 
.2655? 
.  27864 
.29151 
.30419 

!347 
1326 

1305 
1287 
I26§ 

i 

7 
8 

9 

10 

ii 

12 
13 
H 

4.62642 
.  70926 
•78478 
.85431 
.91868 

9151 
8278 
7558 
6953 
6437 

4.62642 

.70921 
.78478 
.85431 
.91868 

9151 
8279 
755? 
6952 

6437 

6.31  666 
.32892 
•34107 

.35305 
.36487 

1250 

1232 

I2IJ 

1198 
Il82 

T  T  AA 

6.31669 
.32901 
.34116 
•353^5 
•  36497 

1250 
1232 
1215 

1  198 
1182 

T  T  f\f^ 

10 

ii 

12 
13 
H 

;i 

17 

18 

19 

4.97866 
5.03466 
.08732 

.I369S 
•18393 

5992 

56°! 

5266 
4964 
4696 

4.97861 

5.03465 
.08732 
.13697 
•18393 

5993 
-5605 
5266 
4964 
4696 

A  A  rf, 

6.37653 
.38803 

•39938 
.41059 
.42165 

I  IOO 

1150 
1135 

II2I 

1106 

6.37663 
.38814 

•39949 
.41076 

.42177 

I  IOO 

1151 
1135 

II2I 
JlOg 

15 

16 

17 
18 

19 

20 

21 

22 
23 
24 

5.22848 
.  27086 
•3II26 
.34987 
.38684 

4455 
4238 
4046 
3861 

-3697 

,,  -  .  P 

5  .  22849 

.27087 
.3112? 

.34988 

.38685 

4450 
4238 
4046 
386T 
3697 

6.43258 
•44337 
.45403 
.46455 
.47496 

1093 
I07§ 
1066 
1052 
1046 

6.43270 
.44349 
.45415 
.46463 

.47509 

1093 
I07§ 
1066 

1053 
1040 

20 

21 

22 
23 
24 

25 
26 

27 
28 
29 

5.42230 

•45636 
.48915 

•  52073 
.55121 

3545 
3406 
3278 
3158 
3048 

5.42231 
.45638 

.48916 

.52075 

•55^3 

3545 
3407 
3278 

3159 
3048 

6.48524 
•49539 
.  50544 
•51536 
.52518 

1015 
1004 
992 
981 

6.48537 

•49553 
•5055? 
.51556 

•52532 

IO28 

1016 
1004 

993 
982 

3 

27 
28 
29 

30 

3i 
32 
33 
34 

5  .  58066 
.60914 
.63672 
.66344 
.6893? 

2944 
2843 

2£5? 
2672 

2593 

•7  C  T  8 

5  .  58068 
.60916 
•63674 
•66345 
.  68940 

2945 
2848 

2758 
2672 

2593 

-r  Tfi 

6.53488 
•  54448 
•55397 
.  56336 
.57265 

970 
960 
949 
939 
929 

6.53503 
•  54463 
•55413 
•56352 
.57281 

97° 
960 

950 

939 
929 

30 

3i 
32 
33 
34 

LO\O  t^OO  ON 
CO  CO  CO  CO  CO 

5-7H55 
.73902 
.76282 

•78598 
.80854 

2447 
2379 

23»6 
2256 

5.7I45? 
.73904 
.76284 
.78601 
.8085? 

2517 
2447 
2380 
2316 

2255 

6.58184 
•  59093 
•  59993 
.60884 
.61765 

919 
909 
900 
891 
882 

o-a 

6.58201 
.59116 
.60011 
.  60902 
.61784 

919 

909 
906 
891 
882 

Q*7  1 

ii 

37 
38 
39 

40 

4i 
42 

43 

44 

5.83053 

.85198 
.87291 

.89335 
•91332 

2199 

2145 
2093 
2044 
1996 

5-83056 
.85201 
.87295 
•89338 
.91335 

2199 
2145 

2093 
2043 

1997 

6.62639 
•63503 
.64359 
.65205 
.66045 

«72 
864 
855 
84? 

839 
o_  ¥ 

6.6265? 
.63522 
.64378 
.65226 
.66065 

073 

864 

856 

848 

839 

8  •? 

40 

4i 
42 

43 
44 

4I 
46 

47 
48 

49 

5.93284 

•95193 
.97061 
5.98890 
6.00686 

1952 
1909 
1868 
1829 
1790 

5.93288 

•95197 
•97065 
5.98894 
6.00685 

1952 
1909 
1868 
1829 
1791 

6.66875 
.67700 
.68515 
.69323 
.70124 

831 
823 

8i5 
808 
806 

6.66897 
.67726 
.68535 

•69345 
.70145 

31 
823 
816 
8o§ 
806 

45 
46 
47 
48 

49 

50 

5i 
52 
53 
54 

6.02435 

.04155 
.05842 

.07496 

.  09  1  20 

J755 
1720 
1685 
1654 
1623 

6  .  02440 
.04160 
.05847 
.07501 
.09125 

J755 
1720 
1687 
1654 
1623 

6.70917 
.71703 
.72482 

.73254 
.74019 

793 
786 

779 
772 
765 

T  ro 

6.70939 
.71725 
.72505 
•7327? 
.74043 

794 
786 

779 
772 
765 

50 

5i 

52 
53 
54 

55 
56 

11 

59 

6.  10714 

.12279 
.13816 

•15327 
.16811 

J594 
1565 

1537 
1511 
1484 
1460 

6.10719 
.12284 
.13822 

.15333 
.16818 

J594 
1565 

1537 

J5" 

1485 
1460 

6.7477? 
•75529 
.76275 

.77014 

•7774? 

758 
752 
745 
739 
733 

72A 

6  .  74802 

•75554 
.  76306 
•  77040 

•77773 

759 

752 

746 
739 
733 

727 

55 
56 

57 
58 
59 

60 

6.18271 

6.18278 

6.78474 

/•'O 

6.78506 

1*1 

00 

Log.  Vers. 

z> 

Log.  Exsec. 

7> 

Loer.  Vers. 

J> 

Ijoer.  Exsec.  1 

1) 

394 


TABLE   VIII.— LOGARITHMIC   VERSED   SINES   AND    EXTERNAL   SECANTS. 

2°  3° 


I 

Log.  Vers. 

D 

Log.  Exsec. 

Z>  | 

Log.  Vers. 

Z> 

I."-'.  Exsec. 

J> 

| 

0 

2 

3 

4 

6.78474 
.79195 
.79909 

.80613 
.81322 

721 
714 
709 
703 
£rv? 

6.78506 
.79221 

•79937  • 
.80646 
.81350 

721 
715 
70§ 

703 
zr_.o 

7.13687 
.14168 
.14645 
.15122 
•15595 

481 

478 
475 
473 

7-13746 
.14228 

.14707 
.15183 
.15657 

48l 

479 
476 
474 

0 

i 

2 

3 

4 

6 

8 
9 

6.820I§ 
.827lt 

.83398 
.84079 

•84755 

O97 
692, 
686 

68  1 
676 

6.82048 
'.82746 
.8342? 
.84109 
.84785 

090 
692 
687 
682 
676 
/r—c 

7.16066 

.16534 
.17000 

.17463 
.17923 

470 
468 
466 

463 
466 

7.16129 
.16598 
.17064 
.17528 
.17989 

47? 
469 
466 
464 
461 

I 

7 
8 

9 

10 
ii 

12 

13 
14 

6.85425 
.86091 

.8675? 
.87407 
.88057 

665 
666 

655 
656 
<  .< 

6-85457 
.86123 
.86783 

.87439 
.88096 

O7I 

666 
666 
656 
651 
f..? 

7.18382 
.1883? 
.19291 
.19742 
.2OI9I 

458 
455 
453 
45? 
448 

7.18448 
.18905 

•19359 
.  1  98  1  1 
.20260 

459 
456 
454 
452 
449 

10 

ii 

12 
13 
H 

15 

16 

17 
18 

19 

6.88703 

.89344 
.89980 
.90612 
.91239 

\  ON  ON  ON  ON  C 
J  N)  U>  U>  4-  4- 
1)  ^1)  1-1)  ON  «->  C 

6.88737 
.89378 
.90015 
.90647 
.91275 

G46 
64? 
636 
632 
628 

/:•>-> 

7.20637 
.2I08T 
.21523 
.21963 
.22400 

446 
444 
442 
440 
437 

At? 

7.2070? 
.21152 
.21595 
•22035 

.22473 

447 
445 
442 
446 
438 

A  lf\ 

15 

16 

17 
18 

19 

20 

21 

22 
23 
24 

6.91862 
.92480 
.93093 
.93703 

•94308 

618 

613 
609 
605 

£r»r 

6.91898 
.92515 
•93I3i 
•93741 

•94346 

023 

6i§ 
614 
610 
605 

Ar»T 

7.22836 
.23269 
.23700 
.24129 
•24555 

43} 
433 
43i 
429 
426 

7.22909 
•23343 
-23775 
.  24204 
•24632 

43& 

434 
43? 
429 
42? 

jf»9 

20 

21 

22 

23 

24 

25 
26 
27 
28 
29 

6.94909 

•955o£ 
.96099 
.96688 
.97272 

597 
592 

589 
584 

,Q  , 

6.94948 

•95545 
.96139 
.96728 
.97313 

597 
593 
589 
585 

-Of 

7  .  24980 
.25402 
•25823 
.26241 
.26658 

424 
422 
426 
4i§ 
4'6 

7.25057 
.25486 
.25902 
.26321 
.26738 

425 
423 

421 

419 
417 

25 
26 

27 
28 

29 

30 

3i 
32 
33 
34 

6.97853 
.98436 
99004 

6-99573 
7-00139 

551 
577 
573 
56§ 
565 

-£.~ 

6-97895 
.98472 

•99046 
6.99616 
7.00182 

S81 

577 
574 
570 
566 

P(L  _ 

7.27072 

•27485 
.27895 
.28304 
.28711 

414 
412 
416 
409 
405 

7.27153 
.27567 
.27978 
.28387 
.28795 

415 

413 
41? 

409 
40? 

.  _a 

30 

3i 

32 
33 
34 

35 
36 

% 

39 

7.00701 
.01259 
.01814 
.02366 
.02914 

502 

558 
555 
55? 
548 

7-00745 
.01304 
.01860 
.02412 
.02966 

563 

559 
555 
552 
548 

7.29116 

•29518 
.29919 

•30319 
.30716 

4°5 
402 
401 
399 
39? 

--.a 

7.29200 
.29604 
.30005 
.30405 
.  30804 

405 
404 
402 
400 
398 

11 
% 

39 

40 

4i 
42 
43 
44 

7-03458 
.03999 

•04537 
.05071 
.05603 

544 
54i 
53? 
534 
53? 

rtn 

7.03505 
.04047 

.04585 
.05126 
.05652 

545 
54? 
538 
535 
53? 

7.31112 

•3i5o5 
.31897 
.32288 
•32675 

395 
393 
392 
390 
388 

_op 

7.31201 
.31595 
.31988 
.32379 
.32768 

396 
394 
393 
39i 
389 

-OO 

40 

4i 
42 

43 
44 

45 
46 

47 
48 

49 

7.06136 
.06655 
.07177 
.07695 
.08211 

527 
525 
52I 

518 
5i5 

7.06186 
.06706 
.07228 

•0774? 
.08263 

r-»o 

S  § 
525 

522 

5I? 
516 

7-33063 
•33448 
.3383? 
.34213 
•34593 

386 

385 
383 
382 
380 

__0 

7-33156 
•33542 
-33926 
.34309 
.34689 

300 

385 
384 
382 
386 

41 
46 

47 
48 
49 

50 

51 
52 
53 
54 

7.08723 
.09232 

.09739 
.  10242 
•  10743 

512 

509 
506 
503 
506 

4r.a 

7.08775 
.09286 

•09793 
.  10297 
•  10798 

^3 
509 
507 
503 
501 

Af\R 

7.3497? 
•35348 
.35723 
.36097 

.36468 

378 
377 
375 
373 
37? 

7.35069 

.35446 
.35822 

•36196 
.36569 

379 
37? 
376 
374 
373 

50 

5i 
52 
53 

54 

55 
56 

H 

59 

7.11  246 
.1^735 

.1222? 

•I27I6 
.13203 

497 
495 
492 
489 
486 
484. 

7.11297 
.11792 
.12285 
.12775 
.  13262 

498 
495 
493 
490 
487 
A.83. 

7-36839 
.37207 

•37574 
.37940 
•38304 

32~ 
368 

367 
366 
364 
^62 

7.36946 
•373'o 
.37678 
.38044 
.38409 

37' 
369 
368 

366 
365 
363 

11 
H 

59 

00 

7.13687 

7  •  i  3746 

7-38667 

7.38773 

owj 

60 

Loar.  Vers. 

7) 

Los.  Exseo. 

D 

Los.  Vers. 

j> 

Loe.  Kx««»r. 

j> 

' 

395 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL   SECANTS. 

4°  5° 


/ 

Log.  Vers. 

J) 

Log.  Exsec. 

D 

Log.  Vers. 

D 

Log.  Exsec. 

J> 

; 

P. 

P. 

0 

I 

2 

3 

4 

7.38667 
.39028 
.3938? 

•3974? 
.40102 

361 

359 
358 
356 

7.38773 
.39134 
.39495 
.39854 
.  402  1  1 

36T 
366 

359 
357 
~ff. 

7.58039 
.58328 

.58615 
.  58902 

.59l8§ 

289 
28? 
287 
286 

,,0- 

7  .  58204 

.58494 
.58783 
.59071 

•59358 

290 
289 
288 
287 
oP.2 

0 

i 

2 

3 

4 

6 

7 

360 

36.0 
42.0 

350 

35-o 

40.§ 

340 

34.o 
39-6 

! 

7 
8 

9 

7.40457 
.40816 
.41163 

.41513 
.41863 

355 
353 
352 
350 
349 

~  .0 

7.4056? 
.40922 
.41275 
.41627 

.41977 

35° 
354 
353 
352 
350 

7-59473 
.59758 
.60041 
.60323 
.60604 

255 
284 
283 
282 
28l 
_o  ~ 

7.59645 
•  59930 
.60214 
.60498 
.60786 

205 
285 
284 
283 
282 
_oc 

.1 

8 
9 

8 
9 
10 

20 
30 
40 
50 

48.0 

54-o 
60.0 

120.0 
iSo.O 
240.0 
300.0 

46-6 
Si.S 
58.§ 
n6.g 
I75-o 
233.§ 
291-6 

45-3 
51-0 
56-6 
"3>3 
170.0 
226.6 
283.3 

10 

ii 

12 
13 

H 

7.42211 

.4255? 
.42903 

.4324§ 
.43589 

34° 
346 
345 
343 
342 
_  .  « 

7.42326 
.42673 
.43019 
.43364 

.43708 

349 
34? 
346 
345 
343 

7.60885 
.6ll64 

.61443 
.61721 
.61998 

2oO 

27§ 
279 

27? 
277 

7.61062 
.61342 
.61622 
.61901 
.62179 

251 
286 
280 
279 

278 

10 

ii 

12 
13 
H 

6 
I 

330 

320 

32.0 
37-3 

310 

g-s 

36.1 

15 

16 

17 
18 

19 

7.43930 
.44270 
.44603 
.44946 
.45281 

34i 
33§ 
338 
33? 
335 

Tjli 

7.44050 
.44390 
.44730 
.45068 

.4540=5 

342 
340 
339 
338 
337 

7.62274 
•62549 
.62823 

.63096 
.63369 

276 
275 
274 
273 
272 

7.62455 
.62733 
.63008 
.63282 
.63556 

277 

276 
275 

274 

274 

15 

16 

17 
18 

19 

9 
10 
20 
30 
40 
So 

49.5 

55-° 

IIO.O 

165.0 
220.  o 

275.0 

42-6 
48.0 

53'3 
io6.g 
160.0 
213-3 
266.g 

46.5 

S*-Jj 

103.3 
155-0 
206.6 
258.3 

20 

21 

22 
23 

24 

7.45616 

•45949 
.46281 
.46612 
.46941 

334 
333 
332 
333 
329 

__g 

7.45740 
.46075 
.4640? 

.46739 
.47070 

335 
334 
332 
332 
333 

7.63641 
.63911 
.64181 

.64451 
.64719 

272 
270 
270 
26§ 
268 

7.63829 
.64101 
.64372 
.64643 
.64912 

273 
272 
271 
276 
269 

20 

21 

22 
23 

24 

0 

7 
8 

300 

30.0 

35.0 
40.0 

290 

29.0 

38-6 

280 

28.0 

32-6 

37-3 

25 
26 

27 
28 
29 

7.47270 

•47597 
.47922 

.48247 
.48576 

328 
327 
32§ 
32* 
323 

7-47399 
.47727 
.48054 

.48379 
.48703 

3^9 
328 
327 
325 
324 

7.64986 
.65253 
.65519 
.65784 

.66043 

267 
26g 
266 
265 
264 
_/r 

7.65181 

.65449 
.65716 
.65982 
.6624? 

268 
267 
266 
265 

ryfiA 

25 
26 

27 
28 
29 

9 

10 

20 

30 

4° 
SO 

45.0 

50.0 

IOO.O 

150.0 

2OO.O 
250.0 

11 

145.0 
193-3 
241.6 

42-0 
46.6 
93-3 
140.0 
iE6.g 
233  3 

30 

3i 
32 
33 
34 

7.48892 
.49213 
•49533 
.49852 
.50169 

322 
321 
320 
3i§ 
3i? 

->TA 

7.49026 
•49348 

.49989 
.5030? 

323 
322 
321 
319 
3i§ 

7.6631! 

.66574 
.66836 
.67097 
.6735? 

^ 
263 

26l 
26l 
266 

7.66512 
.66776 
.67039 
.67301 
.67562 

264 

263 

262 
26! 
~f.. 

80 

3i 

32 
33 
34 

6 

7 
8 

270 

27.0 

31-5 
36.0 

260 

26.0 
30.3 
34-6 

250 

25.0 
29.1 
33-3 

P 

37 
38 
39 

7.50485 
.50806 

.5iii4 
•  5H2? 
.51739 

316 
315 
3*4 

3'3 
31? 

7.50624 

•50941 
.51256 

.51569 
.51882 

3*7 
3i6 
315 
3i3 
3i3 

»  T  7 

7.67617 

.67875 
.68133 
.68396 
.68647 

259 

258 
258 
257 
256 

7-67823 
.68083 
.68342 
.68601 
.68853 

260 
259 

258 
25*? 

35 
36 
37 
38 
39 

y 

10 

20 
30 
40 
50 

40-5 
45-0 
90.0 

I3S'° 
180.0 
225.0 

39>° 
43-3 
8-3-6 
130.0 
173.3 
216-6 

37-5 
41-6 
83.3 
125.0 
166.6 
208.3 

40 

41 

42 
43 

44 

7.52050 

.52359 
.5266? 

.52975 
.53281 

311 
309 
308 

30? 
306 

•50? 

7.52194 
.52504 
.52814 
.53122 

.53429 

31] 

310 

309 
308 
307 

__/r 

7.68902 

.6915? 
.69411 
.69665 
.6991? 

'^i>b 
255 
254 

253 
252 

7.69115 
.69371 
.69627 
.69881 
.70135 

257 
256 

255 
254 

254 

40 

4i 
42 
43 
44 

6 

7 
8 

240 

24.0 

28.0 
32.0 

230 

23'° 

26.5 

3°.  6 

220 

22.0 

25-6 

29.3 

11 

47 
48 

49 

7.53586 
.53896 
.54193 
•54495 
.  54796 

3°5 
304 
303 
302 

300 

7.53735 
.54041 
.54345 
.  54648 
.54950 

3°6 
305 
304 
303 
302 

7.70169 
.70421 
.70671 
.70921 
.71170 

2^2 
25I 
250 
250 
249 

7.70388 
.70641 
.70893 
.7H44 
.71394 

253 
252 
252 
251 
250 

45 
46 
47 
48 
49 

10 

20 
3° 
40 
5o 

40.0 
80.0 

I2O.O 

160.0 
200.  o 

1 

76.6 
115.0 

153-3 
191-6 

i  * 

36.6 

73-3 

IIO.O 

146.6 
'83.  3 

50 

5i 

52 
53 
54 

7.55096 
•55395 
.55692 
.55989 
.56285 

3°° 
299 
29? 
297 

295 

7.55251 
.55550 
.55849 
•5614? 
.  56444 

301 
29§ 

299 
298 

296 

7.7Hi§ 
.71666 

.71913 
.72159 
.72404 

248 
24? 
247 
246 

245 

7.71644 
.71892 
.72141 

.72388 
.72635 

250 
248 
248 
24? 
246 

50 

5i 
52 
53 
54 

6 

I 

0 

210 

21  .O 

24-5 
28.0 

3x-5 

20O 

20.  o 

3:| 

30.0 

I9O 

19.0 

22.1 

3:1 

55 
56 
57 
58 
59 

7.56580 
•56873 
.57166 
.57458 
•  57749 

^ 
293 
293 
292 
296 

2QO 

7.56740 
.57035 
.57329 
.57621 

.57913 

29O 

295 
294 
292 
292 

2QI 

7.72649 
.72893 
.73137 
•73379 
.73621 

243 
244 

243 
242 

242 

->jT 

7.72881 
•73I26 
•73371 
.736i5 
.73859 

246 
245 
245 
244 
243 

2A2 

II 

57 
58 
59 

10 
20 
30 
40 
50 

35-0 
70.0 
105.0 
140.0 

I75-° 

11.1 

IOO.O 

I33-3 
i66.g 

a 

95.0 
126.6 
158.1 

60 

7.58039 

7  .  58204 

^yi 

7.73863 

7.7410! 

00 

LOST.  Vers. 

T> 

Log.  Exsec. 

J> 

Loe.  Vers. 

7> 

Lojr.  Exsec. 

j> 

f 

I'. 

396 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

6°  7° 


' 

LOST.  Verg. 

J> 

Log.  Kxseo. 

J> 

Log.  Vers. 

D 

,og.  Kxser. 

Z> 

P. 

0 

I 

2 

3 
4 

7-73863 
.74104 
-74344 
.74583 
.74822 

241 
240 
239 
239 

_  _0 

7.74IOI 

•74343 

.74585 
.74826 
.75066 

242 
24I 
241 
240 

7.87238 
.87444 
,  .87650 

.87855 
.88060 

205 

205 
205 

204 

7.87563 
.8777' 
.87978 
.88185 
.88391 

208 
207 
207 

205 

1/-\f\ 

0 

2 

3 

4 

6 
7 
8 
9 

180 

18  o 

21  . 
24- 
27- 

?, 

i.t 

1.2 

1.4 

9 

0.9 

I.O 
I  .2 

'•3 

I 

7 
8 

9 

7.75060 
-75297 
•75534 
•75776 
.76006 

230 

237 

236 
236 
235 

7.75305 

•75544 
.75782 
.76019 
.76255 

239 
239 
238 
237 

237 
~~f. 

7.88264 
.88468 
.88672 
.88875 
.8907? 

204 
204 

203 
203 

202 

7.88597 
.88803 
.89008 
.89212 
.89416 

2OO 
205 
205 
204 
204 
ir\5 

6 

8 
9 

xo 

20 

3° 
40 
50 

3°- 
60. 
90. 

120.0 

150.0 

i  6 
3-1 
4-7 
6.1 

7-9 

i-5 
3-° 
4-5 
6.0 
7-5 

10 

ii 

12 

13 
14 

7.76246 
.76475 
.  76703 
.76941 
-77I73 

234 
234 
233 
233 
232 

.7-76492 
•76728 
.76963 
.7719? 
•77431 

230 
235 

III 

233 

7.89279 
.89481 
.89682 
.89882 
.90082 

201 
201 
200 
200 

7.89620 
.89823 
.90025 
.  90228 
.90429 

203 
203 
202 
202 
201 

10 

ii 

12 
13 
U 

6 

8 

9 

10 

8 

0-8 

1.0 

I.I 

1.3 
1.4 

8 

0.8 
0.9 

I.O 
1.2 
I.§ 

? 

o  7 
0.9 

X  O 

I.I 

1  .2 

:i 

17 

18 
19 

7.77405 
.77636 
.77867 
.78097 
-78326 

232 
231 
236 
230 
229 
__  ~ 

7.77664 
.77897 
•78128 
.78360 
.78596 

233 
232 
231 
231 
236 

7.90282 
.90481 
.90680 
.90873 
.91076 

199 
199 

198 
198 
197 

T  r\fj 

7  -  90636 
.90831 
.91032 
.91231 

•9I431 

201 
200 
I99 
I99 

15 

16 

17 
18 

19 

3» 
40 
5" 

4.2 
5-6 
7-1 

7 

4-o 

1:1 

g 

3-7 

5«° 

6.3 

6 

20 

21 

22 

7.78554 
•78783 
.79010 

228 
22g 
227 
227 

7.78826 
.79050 
.79279 

230 
22§ 
229 
228 

7.91273 
.91470 
.91667 

o/r  - 

197 
197 

196 
196 

7.91630 
.91823 
.92027 

199 

198 
198 
IQ7 

•20 

21 

22 

6 

87 
9 

0.7 

0.8 
0.9 

I.O 

0.<j 

°-z 

0-8 

I.O 

0.6 
o-7 
0.8 
0.9 

23 
24 

•79237 
•  79463 

226 

.79507 
•79735 

228 

.91863 
.92058 

195 

.92224 
.92421 

197 

23 
24 

20 

3° 

2-3 
3-5 

2.1 
3-2 

2.0 

3-o 

25 
26 

27 
28 
29 

7.79689 
.79914 
•8o«38 
.80362 
.80586 

?25 

225 
224 
224 
223 

T>?> 

7.79962 
.80183 
.80414 
.80639 
.  80864 

227 
226 
226 
225 
225 

7,92253 

•92448 
.92642 

.92836 
.9102§ 

J95 
195 
194 
194 

193 

7.92618 
.92815 
•  930  1  6 
.93206 
•93401 

197 
J96 

IQ5 

195 
195 

11 

27 
28 
29 

40 
So 

6 
7 

J:i 

s 

°-5 
°-l 

4-3 
5-4 

5 

0.5 

0.6 

4-0 
5.0 

4 

0.4 
0.5 

30 

3i 
32 
33 
34 

7.80803 
.81031 
.81252 

.81473 
.81694 

222 
222 
221 
221 
220 

7.81088 
•81312 

.81535 
.81758 
.81980 

224 
224 
223 
222 

222 

7.93222 

.93415 
.9360? 

•93799 
•93996 

193 
192 
192 
191 
191 

7.93596 
.93790 

•93984 
•94177 
•94370 

J95 
194 
194 
193 
193 

30 

31 
32 
33 

34 

8 
9 
10 

20 

3° 
40 
So 

o  7 
0.8 
0.9 
».| 

ri 

4.6 

0.5 
0.7 

°-i 

'•6 

3-1 
4-i 

0.6 

°-z 

0.7 
T-5 

2.2 

3-o 

3  7 

35 
36 

3 

39 

7.81914 
.82133 
.82352 
-82576 
.82788 

2  2O 
219 
219 

218 
21? 

7.82201 
.82422 
.82642 
.82862 
.83081 

221 
221 
220 
2I§ 
2I9 

7.94181 
•94371 
.94561 

•94751 
.94940 

jyo 
196 
190 
189 
189 

7.94562 
-94754 
•94946 
.9513? 
•95328 

192 
192 
192 
191 
191 

35 
S6 
37 
38 
39 

6 

7 
8 
9 

4 

°-l 
0.4 

0.5 

0.6 
o.2 

3 

°.§ 

0-4 
0.4 
o-S 
0.6 

3 

•3 
•3 
•  4 

40 

41 
42 
43 

44 

7.83005 
.83222 
.83438 
•83653 
•83863 

217 
217 

216 

2I§ 

2,5 

7.83300 
.83518 

.83735 
.83952 
.84169 

219 

218 

21? 
217 

215 

_  _/; 

7.95129 
•9531? 
.95505 
.95693 
.95880 

189 

i«8 
18? 
1  88 
187 

T  SP 

7-95519 
•95709 
•95898 
.96088 
.96275 

196 
190 
189 
189 
i8g 

T°.O 

40 

41 
42 

43 
44 

20 

3° 
4° 
50 

i-3 

2.O 

2-6 
3-3 

2 

i.i 

1:1 

2.9 

2 

.0 

•5 

.0 

-5 

I 

45 
46 
47 
48 

49 

7.84083 
.84297 
.84516 
.84723 
.8493? 

214 
214 
213 

213 
212 

7.84385 
.84606 
.84815 
.85030 
•85243 

2IO 
215 
215 
214 
213 

7.96o6g 
.96253 

.96439 
.96624 
.96809 

186 
186 

1  86 
i8§ 
185 

T  Q? 

7-96465 
•96653 
.96841 
.97028 
•97215 

158 

1  88 
188 
187 
187 

i  R? 

45 
46 
47 
48 

49 

6 

7 
8 

9 

10 

20 

30 

0.2 

°-3 
0-3 
0.4 
0.4 

0  -8 

I  .2 

O.2 
0.2 
O  2 

°-a 

°'3 
°-6 

I.O 

.2 
.2 

.2 
.2 

•  S 

.7 

50 

5» 

52 
53 
54 

7-85147 
•85359 
.85570 
.85780 
.85990 

212 
211 
211 
210 
210 
">OO 

7.85457 
.85670 
.85882 
.86094 
.86305 

<•»  T  0 
213 

213 

212 
211 
211 
21  1 

7.96994 
•97178 
.97362 
.97546 
.97729 

104 

184 
184 
183 
183 

18^ 

7.97401 
.9758? 
•97773 
•97958 
-98143 

106 

1  86 

185 
185 

184 

1  81 

50 

5i 
52 
53 
54 

40 
50 

6 

7 

'•6 

2.1 

I 
O.I 
O.I 

••I 
»-6 

c 

0. 

o. 

.0 

.2 

5 
6 

55 
56 

% 

59 

7.86199 
.  86408 
.86616 
.86824 
.87031 

4\jy 

209 
208 
208 

20? 
20A 

7.86515 
.8672g 

•86935 
.87146 

.87354 

210 
210 
20§ 
208 
''OS 

7.97912 
.98094 
.98276 
.98458 
.98639 

IOJ 

182 
182 
182 

181 
181 

7.9832? 
.98512 
.98695 
.98879 
.99062 

184 
183 
183 
183 
18^ 

55 
56 
57 
58 
59 

8 
9 

10 

20 
3° 
40 
5° 

O.  I 
O.I 
O.I 

03 

°-5 

o.g 
0-8 

o. 

0 

o. 

0. 

o. 
o. 

0. 

8 

i 
i 
i 
i 

3 

4 

60 

7-87238 

•^O 

7.87563 

~\JQ 

7.98820 

7  .  99244 

tiO 

Lot?.  Vers. 

J> 

LOB.  Kxsec. 

J> 

Loe.  Vers. 

j> 

,oir.  Kxsec. 

j> 

I». 

p. 

397 


TABLE   VIII. -LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

8°  9° 


i 

Log.  Vers. 

D 

[joar.  Exsec. 

z> 

Log.  Vers. 

D 

Log.  Exsec. 

j> 

' 

p.  P. 

0 

7.98820 

1  86 

7  .  99244 

182 

8.0903! 

1  66 

8.09569 

162 

0 

2 

3 

4 

.99000 

.  99  i  86 
.99360 
•99539 

1  80 

179 
179 

.99427 
.99609 
.99796 
7-99971 

182 

181 
181 

rRA 

.09192 
.09352 
.09512 
.09671 

1  60 
1  60 
159 

.09732 
.09894 
.  10056 
.  1021? 

162 
162 

161 

I 

2 

3 
4 

180  17 

6   18.0   17 

o  160 

.0   16.0 

6 

8 
9 

7-99718 
7.99897 
8.00075 
.00253 
.00431 

179 
178 
178 
17? 
178 

8.00152 
.00332 
.00512 
.00692 
.00871 

1  86 
1  80 
1  80 

179 

8.09836 
.09989 
.  10148 
.10305 
.  10464 

J59 
159 
158 
158 
158 

8.10378 

•  10539 
.  10700 
.  10866 

.  I  1020 

IOI 

161 
1  66 
1  66 
1  60 
t  f^n 

6 

8 
9 

8     24.0     22 

9   27.0   25 
10   30.0   28 

20    00.0    56 
30    90.0    85 

40  120.0  it3 
50  i  50  .  o  141 

•  8   **'{; 
•6   21.3 
5   =4-o 
•3   26-6 
•6   53-3 
.0   80.0 

•3  Io6-6 
•6  *33.3 

10 

8.00608 

177 

8.01056 

179 

8.10622 

*  57 

8  .  J  i  i  86 

loo 

10 

ii 

12 
13 
14 

.00784 
.00961 
.01137 
.01313 

i'/6 
176 
I76 
176 

.OI229 
.0140? 
.01585 
.01763 

1  78 
178 
178 

17? 

.10779 

•  10936 
.11093 
.11250 

157 
157 
156 

.  i  i  340 

•II499 
.11658 
.11815 

i5§ 

i59 
159 
158 

ii 

12 
13 

6 

7 

ISO 
15.0 
*7«5 

140 

14.0 

•H 

15 

16 

17 
18 

19 

8.01488 
.01663 
.01838 
.02012 
.02186 

i>5 

175 
i75 
174 
174 

8.01940 
.02117 
.02293 
.  02469 
.02645 

177 
177 

176 
176 
175 

I  -7  P 

8.11405 
.11562 
.11718 

.11873 
.  12029 

156 
155 
155 
155 

8.11975 

•12133 
.  12291 

.12448 
.  12605 

T  r  r> 

!58 
158 
158 

15? 

157 

15 
16 

17 
18 

19 

8 
9 
10 
20 

3° 
40 
5° 

20.  o 
22.5 
25  o 
50.0 

75  ° 

1OO.O 

125.0 

18.5 

21  .O 

46.6 
7O.O 

93-3 
Ij6.fi 

20 

8.02359 

173 

8.02820 

!75 

8.12184 

!55 

8.  12762 

!57 

20 

21 
22 
23 

24 

•02533 
.02706 

.02878 
.030:56 

173 
i73 
172 
172 

.02995 
.03176 

•03345 
.03519! 

175 
175 
174 
174 

.12338 

.12492 
.12647 
.12806 

r54 

154 
J54 
153 

.12919 

.13075 

.13232 

J57 
156 

156 

155 
.  r/- 

21 

22 

23 
24 

6 

7 
8 

9,   1 

0.9   o 
I.I   I 

>   8 

9   °-8 
6   i  .0 

2     I.I 

25 
26 
27 
28 
29 

8.03222 
•03394 
•03565 
.03736 
•03905 

172 
171 
171 
171 
176 

8  .  03692 
.03866 
.04039 
.04212 
.04384 

173 
173 
173 
173 
172 

TTO 

8.12954 
.13107 
.  13266 
.13413 
.13565 

]b3 
'53 
'53 
152 
152 

8-13543 
•13698 
.13854 
.14008 
.14163 

156 
155 
155 
154 
154 

25 
26 
27 
28 
29 

9 
10 

2? 

3° 
40 

5° 

1.4  I 

1.6   i 

V  I 

6.3   6 
7-9   7 

3   1-3 
5   i-4 
o   a.jj 
5   4-2 
o   5-6 
5   7-' 

30 

32 
33 

8.04075 
.04245 
.04416 
.04585 

170 
170 
169 
169 

l6q 

8.04556 
.04728 
.04899 
.05076 

172 
'?! 

171 
I  7O 

8.13717 

.13869 
.14021 
.14172 

152 
'52 

I  ci 

8.H31? 
.14471 
.14625 

•14778 

J54 
154 
153 
153 

I  ci 

30 

32 
33 

6 
7 

8 

0.8   o 
0.9   o 

7   7 

7   0.7 
9   0.8 

34 

•04754 

T  (•*• 

.05241  •': 

.H323 

.14932 

34 

8 

i  .0   i 

o   0.9 

35 
36 

39 

8.04922 
.05096 
.05258 
.05426 
•05593 

1  63 
168 
1  68 
16? 
167 

8.05411 
.05581 
.05751 
.05921 
.  06090 

1/U 

170 
170 

169 
169 

i  f^r\ 

8.14474 
.14625 

.H775 
.14925 
.15075 

156 
156 

150 

149 

8.  15085 

.1523? 
•15390 
.15542 
.  i  5694 

J53 
152 
152 
152 
152 

35 
36 

I 

39 

9 
10 

20 

30 
40 

5° 

1.3   i 
4-°   3 

1:1  1 

2     I.I 

5    2.3 
7   3-5 
o   4.5 
2   5-8 

40 

42 

8.05760 
.05925 
.06093 

167 
165 

165 
i  f\f\ 

8.06259 
.0642? 
.06595 

IO9 

1  68 

TAQ 

8.15225 
•15374 
•15523 

150 
149 
149 

8  .  i  5846 

•  *  599? 
.16148 

152 
'5? 

40 

42 

6 

8. 

°-6 

6 

0.6 

43 

.06259 

,/cp 

.06763 

.15672 

149 

.16299 

43 

7 

o  7 

44 

.  06424 

105 

165 
165 
165 

.06931 

I67 
167 
16§ 
T  62 

.15826 

H8 
148 
148 
148 

.  16450 

150 
156 
150 
15° 

44 

8 

9 
10 

20 

3° 
40 

0-8 
i  .0 
1.1 

2  .  I 

3  2 
4-3 

0.8 
0.9 

I.O 
2.O 

3-° 

4.0 

45 
46 
47 

8.06589 
.06754 
.06919 

8.07098 
.07265 
•07431 

8.15968 
.16115 
.16264 

8.16606 
.16756 
.  i  6906 

45 
46 

47 

48 
49 

.07083 
.0724? 

104 

164 

I  A  A 

.07598 
.07764 

105 
1  66 

T  A? 

.  16412 
.16559 

147 

.17050 

.17199 

149 
149 

48 
49 

50 

5-4 

50 

52 
53 
54 

8.07411 

•07575 
.07738 
.07906 
.08063 

104 

163 
163 
162 
162 

T  Ao 

8.07929 
.08095 
.08260 
.08424 
.08589 

165 

165 
164 
l64 
.  £  J 

8.16706 
.16852 
.16999 
.17145 
.17291 

H7 
146 
H6 
146 
146 

M 

8.17349 
.  *  749? 
.17645 

•17795 
.17943 

149 

148 
149 

148 
148 

T  A  P. 

50 

52 
53 
54 

6 

8 

9 

4 

0.6 
o  7 
0.8 

5 

0.6 
0.7 

55 

8.08225 

IO2 
i  AT 

8.08753 

104 

i^5 

8.17437  |Ji 

8.  18091 

I40 

SS 

20 

0.0 

T-6 

56 
57 
58 
59 

.08387 
.08549 
.08710 
.08871 

162 

161 
161 

1  66 

.08917 
.  0908  i 
.09244 
.  09407 

103 
164 
I63 

163 
162 

:$S 

:S3 

.18233 
.18386 

.18533 
.18686 

H7 

lAf. 

56 
57 
58 
59 

30 
40 
50 

2.7 

3-6 
4.6 

2.5 
3-2 
4-1 

00 

8  .  0903  1 

8.09569 

8.18162! 

8.18827 

00 

' 

l/otr.  Vers. 

Loe.  Ex  «eo.   /> 

LOST.  Vors-   7> 

Noe-  Exsec.i  J) 

P.  P. 

398 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 
1O°  11° 


i 

Loe.  Vers.   D 

,nir.  Exsec. 

D 

Los.  Vers.  |  J> 

off.  Exsec.   D 

p. 

P. 

0 
I 

2 

3 
4 

8.l8l62 

.18306 
.18456 

.18594* 
.18738 

144 
144 
144 

8.18827 

.18973 
.19120 
.  1  9266 
.19411 

M6 
146, 
140 

145 

8.2641? 
.26543 
.  26679 
.26816 
.26941 

136 

8.27223 

•27356 
.2749° 
.27623 
.27756 

133 
133 
133 
133 

0 

I 

2 

3 
4 

6 

8.I888I 
.19024 

J43 
143 

8.19557 
.  19702 

'45 

8.27071 
.27201 

130 
I30 

8.27889 
.28021 

132 

6 

130     120 

6     13.0     12.  o 

7 
8 

9 

.19167 
.19452 

142 
142 
142 

.1984? 
.19992 
.20137 

145 
144 

.27331 
.27461 
.27596 

130 
I30 
I2§ 

.28153  ;^ 

.28286   > 
.28418  '3- 

8 
9 

7     15- 
8     17. 
9     »9 

10       21 

5 

i 
I 

14.0 
16.0 
18.0 

20.  O 

10 

8.19594 

142 

8.20281 

144 

8.27719 

129 

8.28550  ;3; 

10 

20     43 

30   65 

.^ 

60  o 

ii 

12 
13 

.19736 
.19878 
.20019 

142 
142 
141 

.20425 
.20569 
.20713 

144 

144 
144 

I  17 

.27849 
.27977 

.28106 

29 
128 
I29 

1  ?Q 

.28681 
.28813 
.28944 

Ji 
131 

ii 

12 
13 

40    86 
50    108 

'. 

3 

8o.O 
IOO.O 

U 

.20166 

141 

.20857 

*43 

.28235 

128 

.29075 

13l 

14 

17 
18 

19 

8  .  20301 
.20442 
.20582 
.20723 
.2086} 

141 
146 
146 
146 
140 

8.21000 

.21143 

.21286 

.21428 
.21571 

*43 
143 
H3 
142 
142 

8.28363 
.  28491 
.28619 
•2874? 
.28875 

128 
128 
128 

12? 

8  .  29206 

•29336 

.29467 

•29597 
.2972? 

136 
136 
136 
130 

II 

17 
18 

19 

6   0.4 
7   0.5 
8   0.6 

4   3 

0.4  0.3 
0.4  0.4 
o.§  0.4 

0.6   0.5 

20 

21 

22 
23 
24 

8.21003  -X 
.21142   39 
.21282   39 

•2I4f  139 
.21560   J" 

8.21713 
.21855 

.21996 
.22138 
.22279 

142 
142 
141 
141 
141 

8.29002 
.29129 
•29256 
.29383 
.29510 

127 

127 
127 
126 
i  ->2 

•*»  il 

:Si  ',i 

.30375  :9 

20 

21 
22 
23 
24 

10   0.7 

20     1.5 
30    2.2 

40   3-0 
5°   3>7 

o.£   0.6 
1.3   i.i 

2.O    1.7 

2'6   2'3 

3-3    2-9 

25 

8.2i698  J38 

8.  22426  ( 

14[ 

8.29636 

6 

8.3050?  IS 

25 

26 
27 
28 
29 

-2I837   1 
.21975   P 
.22113   S 
.2225.  13/ 

.22561  ! 
.22701  • 
.22842 
.22982  ' 

146 
146 
140 

.29763 
.29889 
.30015 
.30146 

-6 
126 
126 
125 

.30633 
.  30762 
.30890 
.31019 

*-y 

I2g 
128 

f>p. 

26 

27 
28 
29 

6    o33 
8    ol 

2 

0.2 

0  3 

30 

32 
33 

34 

8.22389 
.22526 
.22663 
.22800 
.22937 

15° 

'37 
13? 
136 
137 

8.23122! 
.23262 
.23401 
.23546 
.23679 

140 
139 
139 

8  .  30266 
•30391 
•30516 
.30642 

.30766 

i  ->  - 
125 

125 
125 
125 
124 

8.31147 

.31275 
.31402 

.31530 
.3165? 

128 

12? 

12? 
12? 

t/,A 

80 

32 
33 
34 

9    04 
10    05 

SO      I  O 

3°    *  5 

40      20 

5°    2  5 

o  4 
o  4 
o  5 

1  2 

i  6 

2  I 

35 
36 

39 

8.23073 
.23209 

.23346 
.23481 
.2361? 

136 
|36 

8.23813 
.23957 
.24095 
.24234 
.24372 

139 
13S 

13! 
138 

T  -j'? 

8.30891 
.31015 
.31140 
.31264 
.31388 

124 
124 
124 
124 
124 

T  ->  - 

8.31785 
.31912 
.32039 
.32165 
.32292 

127 

127 
127 

126 
126 

P 

37 
38 
39 

2     i 

6    0.2    o.i 

40 

42 
43 

8.23752;  ;*| 

.23888!  35 
.24023   35 
'  24<5?i  3! 

8.24509 
.24647 
.24784 
.24922 

137 
138 
137 
137 
I  37 

8.31511 

•31758 
.31882 

123 
124 
I23 
I2§ 

123 

8.32418 
.32544 
.32676 

•32796 

I26 
126 
126 
126 

!•>§ 

40 

42 
43 

7    o. 
8    o. 
9    0- 

10     0. 
20     0. 

30      I. 

_> 
S 

a 

0 

0.2 
0.2 
0.2 
O.2 

o-7 

44 

.24292'  '34 

.25059 

-  -? 

.32005 

.32922 

T  ~>  - 

44 

40    x. 

I 

I.O 

T   S 

45 
46 

8  .  24426 
.24561 

M4 
134 

8.25195 
.25332 

*36 
136 

8.32128 
.32256 

123 
122 

I  -7  5 

8.3304? 
•33173 

125 

125 

45 

46 

47 
48 
49 

.24695 
.24823 
.24962 

*34 
133 
133 

_  _2 

.25468 
.25604 
.25746 

136 
136 

T  ->A 

.32373 
•32495 
.3261? 

122 
122 

.33298 
•33423 
•3354? 

1^5 

125 
124 

47 
48 
49 

: 

[    6 

50 

52 
53 
54 

8.25095 
.25223 
.25361 

.25494 
.25627 

*33 
133 
133 
132 

133 

8.25876 
.26012 
.2614? 
.26282 
.2641? 

136 
135 
135 
135 
135 

8.32739 
.3286! 

.32983 
•33104 
.33225 

122 
122 
121 
121 
121 

T  -»7 

8.33672 
•33797 
•33921 
.34045 
.34169 

125 
124 
124 
124 
123 

50 

52 
53 
54 

6 

8 
9 

10 
20 

3° 

i 
i 
i 
i 

i 
\ 

? 

0.0 

o.o 

.0 

.1 
.1 

.1 

.2 

P 

57 
58 
59 

8.25759 
.25891 
.  26023 
.26155 

.26286 

132 
132 
132 

8.26552 
.26686 
.26821 
.26955 
.  27089 

134 

134 
134 

134 
134 

I  34. 

8-33347 
.33468 

•33588 
.33709 
.33829 

121 

120 
120 
120 
126 

8.34293 
.34417 
•  34540 
.34663 
.34786 

124 
124 
123 
123 
123 

121 

55 
56 
57 
58 
59 

40 

50 

•3 

0.4 

60 

8.2641? 

8.27223 

8-3395°! 

8  .  34909  i 

60 

:   ' 

Loe.  Vers.   7>  l,oe.  KXSPO.   7> 

Loe.  Vers.   />  lUe.  EXH«T.   J»    ' 

P 

P.         1 

399 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL    SECANTS. 

12°  13° 


Log.  Vers. 

D 

Log.  Exsec. 

Log.  Vers. 

D 

Li»i-r.  Exsec. 

P 

.  P. 

0 

8.33950 

\  "2O 

8  .  34909 

8.40875 

8  .  42002 

I 

0 

I 

.34070 

1  2O 

.35032 

122 

.40985 

1  16 

.42116 

2 

I 

2 

.34190 

.35155 

.41096 

T  in 

.42229 

2 

2 

3 

•34309 

.3527? 

1  22 

.41206 

.42343 

U2 

3 

120 

ng 

118 

4 

.34429 

T  T  A 

.35399 

.41317 

T  t  A 

.42455 

U3 

4 

6 

7 

12.0 
I4.O 

ii.  9 
13.9 

ii.  8 
13.7 

5 

8.34549 

119 

8.35522 

8.4142? 

I  IO 

8.42569 

I!3 

5 

8 

16.0 

iS-7 

6 

8 
9 

.34668 

.34787 
.34906 

.35025 

>  ON  ON  ON  < 

H  I-H  HH  HH  1 

.35644 
•35765 
•3588? 
.36009 

12! 

122 
121 

.4153? 
.4164? 
.41757 
.41867 

IIO 
I0§ 
IIO 

.42682 

.42795 
.42903 
.43021 

)  tO>U>  U>  O- 

6 

7 
8 

9 

:i 

20 
3o 

4o 
50 

20.  o 
40.0 
60.0 
80.0 

100.0 

I9-8 
39-6 
59-5 
79-3 
99.1 

17.7 

39-3 
59-° 
78-6 
98-3 

10 

n 

8.35H3 
.35262 

^8 

8.36130 
.3625? 

121 

12! 

8.41975 
.42086 

.109 

109 

8.43133 
.43246 

112 
112 

10 

ii 

12 
13 

H 

.35386 

•35498 

118 

118 

T  TV 

.36372 

.36493 
.36614 

120 
121 

.42195 
.42304 
.42413 

109 
109 

109 

•43358 
•43470 
.43582 

112 
112 

12 
13 

H 

6 

7 

117 

11.7 
13-6 

116 

ii.  6 

"5 

11.5 
13-4 

15 

8.35734 

117 

TTQ 

8.36734 

8.42522 

109 

8.43694 

112 

IS 

9 

15.6 
!7-5 

15-4 
17-4 

15>3 
17.2 

16 

17 
18 

.35852 
.35969 
.36085 

117 

.36855 
•36975 
•37095 

120 
120 

.42636 

.42739 
.4284? 

109 

1  08 

.43805 
.43917 
•44028 

ii! 
ii! 

16 

17 
18 

10 

20 

30 

40 

19.5 

39  o 
58.5 
78.0 

I9-3 
38.6 
58.0 

77-3 

19. 
38.3 

19 

.  36204 

7 

.42956 

io§ 

T  _o 

.44139 

T  T  T 

19 

50 

97-5 

96.6 

95-8 

20 

8.36321 

H7 

8-37335 

8.43064 

IOo 
TOQ 

8.44251 

III 

20 

21 

.3643? 

•37454 

JI§ 

.43172 

Triq 

.44362 

21 

22 
23 

24 

.36554 
.36671 
.36787 

1  */ 

116 

•37574 
.37693 
.37812 

119 

.43280 
.43388 

•43495 

108 
10? 

•44473 
.44583 
.44694 

I  IO 
110 

22 

23 
24 

6 
7 
8 

114 

11.4 
13-3 
15.2 

M  H  M  M 
0>  10  <1>  00 

112 

II  .2 

13.6 

11 

27 

8.36903 
.37019 
•37135 

116 
116 

8.3793? 
.38050 

•38169 

M^ 
I!? 

8.43603 
•43710 
.438i? 

10? 

107 

10? 

8.44804 

.44915 
•45025 

1  16 

IIO 
IIO 

"I 
26 

27 

9 

10 
20 

17.1 
19.0 

38.0 
57.0 

tn  OJ  w  M 
v,\vj  00  O\ 

>Ln  Os>OO^b> 

16.8 
18.$ 

37-3 
56.0 

28 
29 

•37251 
•37366 

T  T  ? 

.3828? 
.38406 

,  ,Q 

.43924 
.44031 

107 

107 

•45135 
.45245 

109 

28 
29 

40 
50 

95-o 

75-3 
94.1 

74-6 
93.3 

30 

32 
33 
34 

8.37482 

•3759? 
.37712 
•3782? 
•37942 

IJ5 

115 
115 
H5 

T  T  "A 

8.38524 
.38642 
.38760 
.38878 
.38995 

3  OO  OO  OO  CIV  C 

8.44138 
.44245 
.4435? 
.44458 
.44564 

1  05 
107 
105 
log 
1  06 

8.45355 
.45465 
•45574 
.45684 

•45793 

I  IO 
IIO 

109 

109 

109 

30 

32 
33 
34 

j 

III 

ii  .  i 

12.9 
14.8 

IIO 

ii  .0 

10.9 
12.7 
14-5 

£   ~ 

35 
36 
37 
38 
39 

8.38057 
.38171 
.38286 
.38400 
.38514 

114 

114 
114 
114 
114 

8.39H3 
.39230 
.3934? 
•  39464 
.3958? 

H7 
II? 

117 

8  .  44670 

•44776 
.44882 
.44988 
•45°93 

IOO 

105 
105 
1  06 

8.45902 
.4601  ! 
.46126 
.46229 
•46338 

109 
109 

109 

I0§ 

109 

3 

37 
38 
39 

9 

10 

20 

30 

S 

10.6 
18.5 
37-0 
55-5 
74-0 
92.5 

18.3 
36-6 
55-0 
73-3 
91-6 

i8.£ 
36-3 
54-5 
72-6 
90.8 

40 

41 

42 

8.38628 
.38741 
.38855 

114 

113 

114 

8.39698 
.39814 
•39931 

;:| 

8.45199 
.45304 
.45409 

io5 
105 

8.46446 
.46555 
.46663 

108 
1  08 

40 

42 

6 

108 

10.8 

107 

1  06 

43 

44 

.38969 
.39082 

1  13 

.4004? 
.40163 

116 

.455H 
.45619 

105 
105 

•46771 
.46879 

1  08 

43 
44 

7 

S 

12.6 

14.4 
16.2 

12  -5 
14.2 

12.3 
14.1 

45 
46 

8.39195 
.39308 

1  13 

8.40279 
.40395 

1  16 
116 

8.45724 
•45829 

lot, 
104 

IOC 

8.4698? 
•47095 

IOO 
10? 

1  08 

45 
46 

10 

20 

18.0 
36.0 
54-o 

35^1 
53'5 

15-9 

35-3 
53-° 

47 
48 

.39421 
•39534 

113 

112 

.40511 
.40625 

ii5 

T  T  P 

•45934 
.46038 

IV-o 

104 

•47203 
•47316 

10? 

47 
48 

40 
50 

72.0 
90.0 

g:i 

7°-6 

88.J 

49 

•39646 

.40742 

1  J5 

.46142 

•4741? 

107 

49 

50 

8.39758 

T  l9 

8.4085? 

IZ5 

8  .  46247 

104 

8.47525 

107 

50 

51 

52 
53 
54 

•39871 
.39983 
.40095 
.40207 

112 
112 
112 

.40972 
.4108? 
.41202 
•4I31? 

1  15 

nl 

.46351 
.46455 
•46558 
.46662 

104 

104 

103 

104 

.47632 
•47739 
•47846 
•47953 

107 

10? 

107 
105 

52 
53 
54 

6 
o 

105 

10.5 

12.2 
14.0 
15.7 

104 

10.4 

12.  I 

'3-8 
15.6 

6 

0.0 

o.o 
0.6 

O.I 

55 
56 
57 
58 
59 

8.40313 
.40430 

.40541 
.40652 
.40764 

III 
Ii! 
Ill 
III 
III 

8.4143! 
.41546 
.41666 

•4177^ 
.41888 

114 
114 
114 
114 
114 

1  1  A. 

8.46766 
.46869 
.46972 
.47076 
.47179 

^Occo  co<co  CO  c 
D  0  0  0  0  C 

8  .  48060 
.48165 
.48273 
.48379 
.48485 

107 
105 
105 
1  06 

log 
1  06 

55 
56 
57 
58 
59 

10 

M 

30 

40 

50 

17-5 

35-o 
52.5 
70.0 
87.5 

34-6 
52.0 

69-3 
86.6 

O.I 

O.I 
O.2 

o-3 
0.4 

60 

8.40875 

8  .  42002 

8.47282 

*~J 

8.4859! 

60 

Lour.  \  ers.  I 

7> 

^og.  Exsec. 

D 

Log.  Vers. 

1) 

,og.  Exsec. 

7> 

'  | 

P. 

AOO 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL   SECANTS. 

14°  15° 


/ 

Log.  Vers.   D 

Log.  Exsec.   7> 

Log.  Vers. 

T> 

,,0*.  Exsec. 

D 

P.  P. 

0 

2 

3 

4 

8.47282 
.47384 
.4748? 
.4759° 
.47692 

102 
103 
102 
102 

8.4859I 
.4869? 
.48803 
.48909 
.49014 

1  06 
1  06 

105 

T/->? 

8.53242 

•53338 
•53434 
•53530 
.53625 

96 

95 

96 

95 

f\? 

8.54748 

54847 

•  54946 
•55045 
.55144 

99 
99 
99 
99 

0 

i 

2 

3 
4 

103   102   IOI  1 

i 

8-47795 
.47897 

I  O2 
102 
I  O2 

8.49120 
.49225 

105 

I05 
105 

8-53721 
•53816 

95 
95 

QC 

8.55243 
•55342 

99 
9? 
98 

0 

6 
I 

10.3 

12.0 
13-7 

10.2 

ii.  9 

13.6 

10.  I 

ii.  8 

7 
8 

9 

•47999 
.48101 
.48203 

102 
102 

T/-V? 

.49331 
.49436 
•49541 

105 
105 

•539II 
.54007 
.54102 

95 
95 

•55441 
•55539 
.55638 

98 
98 

8 
9 

9 
10 

20 

3° 

15-4 
34-3 

15.3 
17.0 

34.0 
51.0 

i6.§ 
33-6 
5°.  5 

10 

ii 

12 
13 
H 

8.48304 
.48406 
.4850? 
.48609 
.48716 

IOI 
TOI 
IOI 
IOI 
IOI 

8.49646 
.49750 
.49855 
.49960 
.50064 

105 
104: 
105 
104 
104 

8.54197 
.54291 

•  54386 
.5448i 
•54575 

95 
94 
95 
94 
94 

r\A 

8-55736 
.55834 
•55933 
.56031 
.56129 

98 

98 

98 
98 
98 

rw 

10 

ii 

12 
13 

4" 
50 

85-8 

85.0 

84.5 

17 

18 

8.4881! 
.48912 
.49013 
.49114 
.49215 

IOI 
IOI 
IOI 
IOO 
IOI 

8.50l6§ 
•  50273 
•  50377 
.50481 

•50585 

104 
104 
104 
104 
104 

8  .  54670 
.  54764 
.54858 
•  54952 
•55046 

94 
94 
94 
94 
94 

8.56225 

•56324 
.  56422 
.56519 
.56617 

97 

98 

97 
97 
97 

15 

16 

17 
18 

19 

6 

- 
8 
9 

IOO 

IO.O 

"•6 
13-3 
15.0 

16.6 

99 

9.9 

"•5 
13-* 

14-8 
16.5 

98 

9.8 
11.4 
13-0 

11:? 

20 

21 
22 
23 
24 

8.49315 

•49516 
.49616 
.49716 

IOO 
IOO 

IOO 
IOO 

8.50688 
.50792 
.50896 

•  5°999 
.51102 

103 
104 
103 
103 
103 

8.55146 

•55234 
.55328 
•55421 
•55515 

94 
93 
94 
93 
93 

8.56714 
.56812 
.56909 
.57006 
•57103 

97 

97 
97 
97 
97 

20 

21 

23 
24 

20 
30 
40 
50 

33-3 
50.0 
66-6 
83.3 

33-0 

49-5 
66.0 
82.5 

3«-6 
49.0 

65.3 

25 
26 

8.49816 
.49916 

IOO 
IOO 
ori 

8.51205 
.51309 

103 
103 

8.55608 
.55701 

93 
93 

8.57200 
•57296 

97 
96 

Q7 

25 
26 

27 
28 
29 

.50015 
.50H5 
.50215 

99 

IOO 

99 

.51412 
.5I5H 
.51617 

103 

102 
103 

•55795 
.55888 

.5598r 

93 

93 
93 

•57393 
•  57490 
•57586 

96 
96 

27 
28 
29 

6 

7 

97 

.  9-7 
"•3 

96 

9.6 

•  II  .2 

95 

9-5 
ii.  i 

30 

8.50314 

99 

8.51720 

T/-*0 
I  O2 

8.56074 

93 
93 

8.57682 

Q6 

30 

9 

12.  § 

14.5 

14.4 

14.2 

32 
33 

34 

.50413 
.50512 
.50611 
.50716 

99 
99 
99 
99 

r»o 

.51822 
.51925 
.  52027 
.52129 

102 
102 
102 

.56165 
•56259 
•56352 
.56444 

~ 
92 
93 
92 

•57779 

;  57875 

•57971 
.5806? 

95 
96 

32 
33 
34 

10 

20 

40 
5° 

i6.i 

32-3 
48-5 
64-6 
So.g 

16.0 
32.0 
48.0 
64.0 
80.0 

15-8 

47-5 
63-3 
79.1 

35 
36 

8  .  50809 
.50908 
.51005 
.51105 

98 
99 
98 
98 

no 

8.5223! 
.52333 
.52435 
•52537 

102 
102 
IOI 

T^? 

8.56536 
.  56629 
.56721 
.56813 

92 

92 
92 

92 

9-7 

8.58163 
.58259 

.58354 
.58456 

5 
96 

95 
96 
o5 

37 

39 

.51203 

98 

nP. 

•5263§  lot 

.56905 

* 

.58546 

yj 

39 

94   93   92 

40 

42 
43 
44 

8.51301 
•51399 
.51497 

•51693 

95 
98 
98 
98 

97 

_Q 

8.52740   \"\ 

•5'84fj  ° 

•  5*943  j  oi 
.53044  IOI 

•53U5'  im 

8.56997 
•  57089 
.57186 
.57272 
.57363 

92 

9? 

8.58641 
.58736 
.58832 
.  58927 
.59022 

95 
95 
95 
95 
95 

40 

42 
43 
44 

i 

9 

10 

20 
3° 

9.4 
10.9 
12.5 
14.1 

3i'l 
47.0 
62.5 

9-3 
10.8 
12.4 
13-9 
I5-5 
31.0 
46.5 
62.0 

9.2 
10.7 

12.2 
13-8 

15  3 
30-6 
46.0 
61.3 

45 
46 

47 
48 

8.51791 
.51883 
.51986 
.  52083 

90 

9? 
9? 
97 

8.53246  \°Q\ 
•53347 
•53448  \™ 
•53548;  100 

8-57455 
.57546 
.5763? 
•57728 

9* 

9* 

9' 

8.59"7 

.59211 

•  59306 
•  594oi 

95 
94 
95 
94 

QA 

45 
46 

47 
48 

50 

77-5 

76-1 

49 

.52186 

97 

.53649!  °? 

' 

•  59495 

yr 

49 

50 

51 

8.52277 
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9V 
97 

8-53749   £ 

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8.57916 
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91 
96 

8.59590 
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94 
94 

QJ. 

50 

6 

91 

10.6 

90 

9.0 
10.5 

6 

o.o 
o.o 

52 
53 
54 

.52471 
.52568 
.52665 

97 
96 

97 

•  53950 
•  54050 
.54150 

IOO 
,00 

.58092 
.58182 
•58273 

90 
96 

•59779 
.59873 
•  59967 

94 
94 

52 
53 

54 

8 
9 
10 

20 

12.  I 

\\\ 

30.3 

12.0 
13.5 

J5  o 
30.0 

o.o 

O.  I 
O.I 
O.I 

55 
56 

59 

8.52761 
.52858 
•52954 
•53050 
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96 
96 
96 

96 
96 
06 

8-  54*50  |£ 

:IS5  ,2 

•  54549  JIQ| 

.  54649  !   ^ 

8.58363 
-58453 
.58544 
.58634 
.58724 

90 

90 
90 
90 

90 
oo 

8.60061 
.60155 
.  60249 
•60342 
•60435 

94 
94 
94 
93 
94 
°3 

55 
56 
57 
58 
59 

3° 
40 
50 

450 
60.6 

75-8 

45.0 
60.0 
75.0 

0.2 

o-3 

0.4 

GO 

8.53242 

8  .  54748 

8.58814 

8.60530 

00  ' 

' 

Loir.  Vers. 

n 

Log.  Kxsec.   T) 

Log.  Vers.   X> 

Log.  Kxsec.   J> 

p.  P. 

401 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL   SECANTS. 

16°  17° 


' 

Log.  Vers 

j> 

Lot;.  Exsec 

D 

Log.  Vers. 

D 

Log.  Exsec. 

D 

' 

p.  P. 

0 

I 

8.58814 
.  58904 

90 

OA 

8.60530 
.60623 

93 

8  .  64043 

.64128 

84 

07 

8.65984 
.66072 

88 

0 

I 

2 

•  58993 

59 

.60715 

93 

.64212 

04 

.66166 

88 

2 

3 

4 

.59083 
.59173 

9° 
89 

OA. 

.60810 
.60903 

3 
93 

_2 

.64295 
.64381 

84 

.66248 
.66336 

88 
8s 

3 

4 

6 

7 

93 

9-3 

10.  g 

92 

9-2 

10.7 

91  1 

9.1 
10.6 

I 

8.59262 
.59351 

89 
89 
OA 

8.60995 
.61089 

93 
93 

8.64465 
.64549 

0  A 

84 
84 

Q  , 

8.66425 
.66512 

8 
8? 

88 

6 

8 
9 

10 

12.4 
13-9 
I5-5 

12.2 
I3.8 

12.  1 
13-6 

IS-1 

7 

.  59441 

59 
0~ 

.6ll8§ 

93 

•64633 

04 

.66606 

88 

7 

20 

31.0 

30-6 

3°-3 

8 
9 

.59530 
.59619 

59 
89 

.61275 
.61368 

92 
93 

.64717 
.64801 

84 

.6668§ 
.66776 

87 

8 
9 

3<> 
40 
50 

46.5 
62.0 

77.5 

46.0 

61.3 
76.6 

45-5 
60.  6 
75-8 

10 

8.59708 

on 

8.61466 

92 

8.64884 

83 

8.66863 

QQ 

10 

ii 

12 

13 

•  59797 
.59886 

•  59974 

09 
8? 
80 

.61553 
.61645 
.61738 

92 

92 

.  64968 
.65052 
.65135 

84 
83 

.66951 
.67039 
.67126 

>0  OO  OO  C 
-D^J  ^J)  C 

ii 

12 
13 

5 

90 

89 

8  o 

88 

8  8 

.60063 

°9 
8g 

.61830 

92 

no 

.65218 

3 

.67213 

^  1 

14 

10.5 

10.4 

10.2 

15 

16 

8.60152 
.60246 

8 

8.  61922 
.62014 

92 
92 

8.65302 
.65385 

83 

8.67301 
.67388 

87 

15 
16 

9 

10 

12.  0 
13-5 
I5.0 

13-3 

II.7 
I3.2 

17 
18 

19 

.60323 
.60417 
.60505 

88 
88 

00 

.62105 
.62198 
.62296 

92 
92 

92 

.65468 
.65551 
-65634 

83 

83 
o- 

.67475 
.67562 
.67649 

87 

87 

17 

18 
19 

20 

3° 
40 

30.0 

45  -o 
60.0 
75-0 

29-6 
44-5 
59-3 
74.1 

29-3 
44.0 

58-6 

73-3 

20 

8.60593 

QQ 

8.62382 

91 

8.6571? 

83 

8.67736 

Q? 

20 

21 

.60681 

88 

.62474 

92 

.65806 

OS 

.67822 

°6 

21 

22 
23 

.  60769 
.60857 

88 

.62565 
.62657 

91 
9? 

•65883 
.65965 

82 

8-7 

.67909 
.67996 

86 

Q? 

22 
23 

6 

87 

8.7 

86 

8.6 

85 

8.5 

24 

.60944 

OS 

•62748 

9^ 

.66048 

°3 

.68082 

°6 

24 

7 

10.  1 

ii  6 

10.0 

11.4 

9.9 

25 

8.61032 

07 

8.62840 

91 

8.66131 

82 

8-7 

8.68169 

Q? 

2$ 

9 

13-5 

12.9 

12.7 

26 

.61119 

pfl 

.62931 

91 

.66213 

OS 

.68255 

°6 

Qf. 

26 

10 

14.5 

28.5 

14.1 
28.3 

27 

.61207 

°/ 

.63022 

oT 

.66295 

OS 

.68341 

Q? 

27 

3° 

43-5 

42.5 

28 
29 

.61294 
.61381 

87 

Q<3 

.63113 
•63204 

91 

•66378 
.66460 

82 

Q_ 

.68428 
.68514 

°6 

86 

Of. 

28 
29 

40 

50 

58.0 
72.5 

57-3 
71-6 

56.6 
7°-8 

80 

32 

8.61469 
.61556 
.61643 

>0  OO  OO  C 

J  *vj  VJ  V 

8.63295 
.63386 
.63477 

2  «  _  <( 

5s  ON  ON  C 

8.66542 
.66624 
.66706 

o2 
82 

82 

8~ 

8.68600 
.68685 
.68772 

86 

85 

Qf. 

30 

32 

84   83   82 

33 
34 

.61730 
.61815 

«/ 

86 

0- 

.6356? 
.63658 

y^ 
90 

.66788 
.66870 

82 

.68858 
.68944 

86 

8p 

33 
34 

6 

I 

8.4 
9.8 

I  I  .2 

»-3 
9-7 

II.  0 

5.2 

9-f 
10.9 

35 

36 

8.61903 
.61990 

07 

8.63748 
.63839 

96 
9° 

8.66951 
.67033 

Ol 

81 

8~ 

8.69029 
.69115 

5 
86 

QP 

3 

9 

10 

20 

14.0 
28.0 

12.4 
13-8 
27-6 

12.3 
13-6 
27-3 

39 

.62075 
.62163 
.62249 

86 
86 

o? 

.63929 
.64019 
.64109 

9° 
96 
90 

.67115 
.67195 

.6727? 

8T 
81 

.69201 
.69285 
.69372 

0  OO  OO  C 
i  vj-ovj-ou 

37 
38 
39 

3" 

40 

5° 

42.0 
56.0 
7O.O 

55-3 
69.1 

41.0 

54-6 
68.3 

10 

41 
42 

8.62336 
.62422 
.62508 

XD  OO  OO  C 
7s  ON  ONO 

'.64289 
.64379 

90 
90 
90 

8-67359 
.67446 
.67521 

ol 

81 

81 

Or 

8.69457 
.69542 
.6962? 

°!> 

85 

85 

40 

42 

6 

81 

8.1 

80 

8.0 

79 

7.9 

43 

.62594 

05 

.64469 

90 

.  67602 

8  T 

.69712 

8E 

43 

7 

9-4 

9-3 

9-2 

44 

.62680 

ofi 

.64559 

9 

.67683 

ol 

.69798 

8  r 

44 

8 
9 

10.8 

12.  1 

10.  g 

12.  O 

10.5 
ii.  g 

45 
46 
47 
48 

8.62766 
.62852 
.62937 
.63023 

OO 

86 
85 
85 

8  .  64649 

.64738 
.64828 
.6491? 

9° 

8.67764 

.67845 
.67926 
.68007 

51 

81 
86 
81 

OA 

8.69883 
.6996? 
.70052 
•7013? 

85 

8| 
85 
85 

45 
46 
47 
48 

10 

20 

30 

4° 

5° 

13-5 
27  0 

40  5 

I3-3 
26-6 
40.0 

8:1 

26^3 
39-5 
52  6 
65-8 

49 

•63108 

89 

.65005 

? 

.6808? 

OS 

.70222 

8~ 

49 

50 

8  .  63  i  94 

5 

8.65096 

8 

8.68168 

OO 
OA 

8  .  70305 

Q7 

50 

52 

.63279 
•63364 

85 

Qc 

.65185 
.65274 

89 

.68248 
.68329 

86 

OA 

.70391 
.70475 

04 

84 
Q7 

5' 
52 

6 

6 

o.o 

53 
54 

.63449 
.63534 

°5 
85 

Qc 

.65363 

•65452 

89 

Q.n 

.68409 
.68489 

80 

.70560 
.70644 

04 

84 
Q. 

53 
54 

7 
8 
9 

o.o 
o.o 

0  I 

55 

8.63619 

55 

8.65541 

59 

86 

8.68569 

OA 

8.70728 

«4 
07 

55 

10 

O.  I 

56 

•63704 

~5 

.65629 

°8 

85 

.68650 

00 

.70813 

04 
QA 

56 

3° 

0.2 

57 

.63789 

8" 

.65718 

8 

.68730 

•  70897 

04 

57 

40 

o-3 

58 
59 

.63874 
.63959 

85 

81 

.65807 
•65895 

85 

.68810 
.68889 

79 
80 

.70981 
.71065 

84 

8/1 

58 
59 

5° 

8  .  64043 

8.65984 

°o 

8.68969 

8.71149 

60 

'    LOST.  Vers.  /> 

joer.  Kxsec.l  J> 

Loar.  \>r«. 

7> 

x>sr.  KXKPC. 

P.  P. 

402 


TABLE  VIIL— LOGARITHMIC  VERSED  SINES  AND  EXTERNAL  SECANTS. 
18°  19° 


/ 

Los.  Vers. 

D 

Log.  Exsec 

j> 

Log.  Vers. 

D 

Log.  Exsec. 

D 

P.  P. 

0 
I 

8  .  68969 

.69049 

79 

8.71149 
.71232 

83 

8.73625 
.73706 

75 

8.76058 
.7613? 

79 

0 

i 

2 

3 

4 

.69129 
.69203 
.69288 

7§ 
79 
79 

.71316 
.71400 

.71484 

83 
84 

8? 

.73775 
.73851 
.73926 

75 

75 
75 

75 

.7621? 
.76297 
•76376 

79 
79 
80 

70 

2 

3 

4 

6 

84 

8.4 

8-3 

82 

8.2 

5 

8.69367 

8.71567 

8  .  74001 

8.76456 

5 

6 

8 
9 

.69446 
.69526 
.69605 
.69684 

9 
79 
79 
79 

.71651 
.71734 
.7181? 
.71901 

833 
83 

83 

Q_ 

.74076 
.74151 
.74226 
.74301 

75 
75 
75 
75 

.76536 
.76615 
.76694 
.76774 

79 
79 
79 
79 

6 

8 
9 

j 

9 

10 
20 
30 

9.8 

II.  2 
12.6 
14.0 
28.0 
42.0 

9-7 
ii.  6 
12.4 
13-8 
27  6 
41  5 

9-5 
10.9 
12.3 
13-6 
27-3 
41.0 

10 

ii 

8.69763 
.69842 

79 
79 

8.71984 
.72067 

83 

8.74376 
•74451 

75 
74 

8.76853 
.76932 

79 
79 

10 

ii 

40 
50 

56.0 
70.0 

55-3 
69.1 

54>  6 
68.3 

12 

.69921 

79 

70 

.72150 

8 

.74526 

75 

7? 

.77011 

79 

7O 

12 

13 

.70000 
.70079 

78 
79 

•70 

.72233 
.72316 

83 

.74606 
.74675 

/4 
74 

.77090 
.77169 

/y 
79 

13 
14 

15 

16 

8.70157 
.70236 

78 
78 

75 

8.72399 
.7248T 

82 

P- 

8.74749 
.74824 

74 
74 

8.77248 
•7732? 

79 
79 

15 

16 

A 

81 

8  i 

80 

8  o 

79 

17 

•70314 

8 

.72564 

°3 

OS 

.74898 

.77406 

78 

7O 

17 

7 

9.4 

9-3 

9.2 

18 
19 

.70393 
.70471 

78 
78 

78 
78 

78 

.72647 
.72729 

82 
82 
82 
82 

•74973 
.75047 

74 
74 

74 
74 

7/1 

.77485 
.77563 

79 

78 
78 
78 

7O 

18 
19 

8 
9 

10 

20 
30 
40 

10.8 
12  i 

13-5 
27.0 
40.5 
54-0 

io.g 

12.  O 

a 

40.0 
53.3 

26.3 

39-5 

20 

21 

8.70550 
.70628 

8.72812 
.72894 

8.75121 
•75195 

8.77642 
.77720 

20 

21 

22 

.70706 

.72977 

•75269 

.77799 

22 

50 

67.5 

66.6 

65.3 

23 

'24 

.70784 
.70862 

78 
78 

«0 

.73059 
.73HI 

82 

P.-T 

•75343 
.7541? 

74 
74 

.77877 
.77956 

78 

23 
24 

25 
26 
27 

8.70946 
.71018 
.71096 

70 

78 

71 

8.73223 
.73306 
.73388 

82 
82 
82 

8.75491 
.75565 
.75639 

74 
73 
73 

8.78034 
.78112 
.78191 

78 
78 

•  78 

78 

25 
26 

27 

78   77   76 

28 
29 

.71174 
.71251 

77 

,-G 

.73470 
•73551 

8! 

.75712 
•75786 

73 
73 

.78269 
.78347 

78 

~Q 

28 
29 

6 
7 
8 

7.8 
9.1 
10.4 

7-7 
9.0 

IO.2 

7-S 

10.  1 

30 

32 
33 
34 

8.71329 

.71406 
.71484 

.7156? 
.71639 

77 

I 

8.73633 
.73715 

•73797 
•73878 
.7396o 

82 
8! 
81 

81 

Of 

8.75860 

.75933 
.76005 
.76080 
•76153 

74 
73 
74 

73 
73 

8.78425 
•78503 
.78581 
.78659 
.78736 

7s 
78 
78 
78 
77 

30 

32 
33 

34 

9 
10 

20 

3° 
40 
5° 

ii.  7 
13.0 
26.0 
39-0 
52.0 
65.0 

"•§ 

12.  g 
25-6 
38.| 

S'.l 

I2'4 

25.3 
38.0 

63^3 

35 
36 
37 

8.71716 

•71793 
.71870 

77 
71 
77 

8  .  74041 
.74123 
.74204 

ol 

81 
81 

0? 

8.76225 
.76300 
.76373 

73 
73 
73 

8.78814 
.78892 
.78969 

77 

71 

7*7 

P 

37 

38 
39 

•71947 
.72024 

77 
77 

.74286 
.74367 

81 

0, 

.76446 
.76519 

73 

73 

.79047 
.79124 

// 

71 

7<7 

38 
39 

6 

75 

7-5 

74 

7-4 

73 

7-3 

40 

8.72IOI 
.72178 

77 
76 

8.74448 
.74529 

81 

Qf 

8.76592 
.76664 

73 
72 

8.79202 
.79279 

77 
77 

7*7 

40 

7 
8 

9 

8.7 

IO.O 
II.  2 

9-8 
ii.  i 

8.5 

9-f 
10.9 

42 
43 

.72255 
.72331 

Ii 
76 

.74616 
•74691 

81 
86 

.7673? 
.76810 

73 
72 

.79357 
•79434 

77 

77 

42 
43 

10 

20 
•3,0 

12.5 
25.0 

37-5 

12.3 
24-6 
37.0 

12.  1 

24-3 
36.5 

44 

.72408 

76 
76 

76 

72 

•74772 

81 
81 
86 
86 

.76883 

72 
72 

72 

7? 

•795II 

71 
77 
77 

77 

44 

40 
50 

50.0 
62.5 

49-3 
6i.6 

60.  § 

11 

47 

8.72485 
.7256? 
.72637 

8.74853 
.74934 
.75014 

8.76955 
.77028 
.77106 

8.79588 
.79665 
.79742 

46 

47 

48 
49 

.72714 
.72796 

o 
76 

76 

76 

.75095 
.75175 

86 

86 
86 

.77173 
.77245 

72 

72 

72 

.79819 
.79896 

77 
76 

77 

48 
49 

6 

72 

7.2 

71 

7.1 

6 

0.6 

50 

8.72865 

8.75256 

8.773^ 

8.79973 

50 

52 
53 

•  72942 
.73018 
.73094 

/u 

76 
76 

76 

•75336 
•75417 
•7549? 

86 
86 

.77390 
.77462 
.77534 

72 

72 

.80050 
.80125 
.80203 

76 

77 

52 
53 

I 

9 
10 

8.4 
9.6 
10.8 

12.  O 

9-4 
io.g 
"•8 

0.6 
0.6 

O.I 
O.I 

54 

.73176 

/u 

76 

76 

75 

•75577 

86 
80 
80 
80 

P,r» 

.77606 

• 

72 
72 
72 

7? 

.80280 

76 
76 
76 
76 
76 

54 

20 

3° 
40 
50 

24.0 
36.0 
48.0 
6O.O 

23-6 
35-5 
47-3 
59-1 

o.i 

0.2 

0.4 

1 

8.73246 
.73322 
•73398 

•73473 

8.75658 
.75738 
.75818 
.75898 

8.77678 
.77750 
.77822 

.77893 

8.80356 
.80433 
.80509 
.80586 

11 

57 
58 

59 

•73549 

7? 

•75978 

80 

.77965 

72 

7T 

.80662 

/u 

76 

59 

00 

8.73625 

8.76058 

8.78037  ' 

8.80738 

60 

' 

Loir.  Vers 

Loir.  Exsec 

Log.  Vers.!  J> 

LOST.  Exsec.   D 

P.  P. 

403 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES   AND    EXTERNAL   SECANTS. 

2O°  21° 


/ 

jog.  Vers. 

r> 

og.  Exsec.   D 

Log.  Vers. 

jOff.  Exsec. 

D 

p.  P. 

f 

8.78037 
.78103 

7? 

8.80738 
.80814 

76 

8.8222§ 
.8229? 

68 
65 

8.85214 

.8528? 

73 

0 

i 

2 

.78180 

7\ 

.80891 

76 

.82366 

8 

6R 

.85366 

l\ 

2 

3 

4 

.78251 
.78323 

n 

7* 

.80967 
.81043 

7° 

76 
76 

76 

.82434 
.82502 

68 
67 

68 

.85433 
.85506 

72 
73 
73 

3 

4 

6 

76 

7.6 

75 

7-5 

74 

7.4 

5 

8.78394 

8.81119 

8.82569 

8.85579 

5 

6 
7 

.78466 
.78537 

71 

.81195 
.81271 

/° 

76 

.8263? 
.82705 

68 

6^7 

.85651 
.85724 

72 

73 

«£ 

6 

7 

7 
8 

9 

8.  g 

TO.  I 

8.7 

IO.O 
II  .2 

8.6 
9-8 
ii.  i 

8 

.78603 

71 

.81346 

6 

.82773 

07 
6R 

.85797 

72 

7? 

8 

0 

I2.g 

12-5 

12.3 

9 

.78679 

71 

.81422 

7  a 

.82841 

fR 

.85869 

72 

70 

9 

9O 

25.3 
38.0 

25.0 

37-5 

24-6 

10 

ii 

8.78756 
.78821 

n 

8.81498 
.81573 

5 

8.82908 
.82976 

67 

6? 

6*7 

8.85942 
.86014 

72 

72 

79 

10 

ii 

40 

5° 

63*1 

5O.O 

62.5 

49-3 
61.6 

12 
13 
H 

.78892 

.78963 
.79034 

76 

.81649 
.81725 
.81806 

75 
75 

T? 

.83043 
.83111 

.83178 

07 
6? 
6? 

/re 

.86087 

.86159 
.86231 

72 
72 
72 

12 
13 
14 

II 

17 

18 
19 

8.79105 
.79175 

•79246 
.79317 
.7938? 

76 

76 
76 

8.81876 
.81951 
.  82025 
.82102 
.82177 

75 
75 
75 
75 
75 

8.83246 

.83313 
•83386 
.8344? 
.83515 

07 
67 

6? 
67 
6? 

6*7 

8.86304 
.86376 

.86448 
.86526 
.86592 

72 
72 
72 
72 
72 

Ii 

17 

18 

19 

6 
7 
8 
9 

10 

73 

7-3 
9*7 

IO.O 
12.1 

72 

7-2 

8.4 
9.6 

10.8 

12.0 

71 

7-1 

9'| 

ii.  § 

20 

21 

22 
23 

24 

8.79458 
•79528 
•79598 
.79669 
.79739 

70 
76 
70 
76 
70 

8.82252 
.82327 
.  82402 
.8247? 
.82552 

75 
75 
75 
75 
74 

8.83582 

.83649 
.83716 

.83783 
•83850 

07 

67 
67 
67 
67 

A? 

8  .  86664 

.86736 
.868o§ 
.86886 
.86952 

72 
72 
72 
72 
71 

20 

21 

22 

23 

24 

20 
JO 

40 
50 

24.3 

4s:| 

60.8 

24.0 
36.0 
48.0 
60.0 

23-6 
35-5 
47-2 
59-1 

25 

8.79809 

7° 

8.82627 

75 

8.83915 

66 

6*7 

8.87024 

72 

25 

26 

.79879 

70 

.  82702 

75 

.83983 

07 

.87095 

26 

27 
28 

•79949 
.80019 

7° 
70 

.82775 
.82851 

74 
75 

.84050 
.84117 

67 
62 

.8716? 
.87239 

72 
71 

27 
28 

6 

70 

7.0 

69 

6.9 

68 

6.8 

29 

.80089 

70 

•  .82926 

4 
•7/r 

.84183 

°6 

.87316 

71 

29 

g 

8.1 
9-3 

8.5 
9.2 

7-9 
9.0 

30 

32 
33 
34 

8.80159 
.80229 
.  80299 
.80369 
•80438 

70 

6§ 
70 
69 

8.83006 
.83075 
•83149 
.83224 

.83298 

74 
74 
74 
74 
74 

8.84250 
.84316 
.84383 
.84449 
.84515 

66 
66 
66 
66 

A2 

8.87382 
•87453 
.87525 
.87596 
.87668 

7i 

30 

32 
33 

34 

9 
10 

20 

3° 

40 

5° 

10.  < 

IX.  g 

23-3 

35  -o 
46-6 
58.  § 

10.3 
"•5 
23-0 
34-5 
46.0 
57-5 

10-2 

•il 

45-3 
56-6 

35 

8.80508 

6n 

8.83373 

74 

8.84582 

05 

62 

8.87739 

7i 

35 

36 
37 

.8057? 
.  80647 

09 

69 

.83447 
.83521 

74 
74 

.84643 
.84714 

°6 

66 

.87816 
.87881 

7i 

36 

37 

38 
39 

.80715 
.  80786 

69 

.83595 
•  83670 

74 
74 

.  84786 
.84845 

66 

.87953 
.  88024 

71 

38 
39 

6 

67 

6.7 

66 

6.6 

65 

6-5 

40 

8.80855 

rS 

8.83744 

74 

8.84912 

66 

8.88095 

71 

40 

7 

7.8 

7-7 

7-6 

41 

.80924 

6n 

.83818 

74 

.84978 

66 

.88166 

71 

41 

Q 

8.9 

IO.O 

9-9 

8. 

9-7 

42 
43 
44 

.80993 
.81063 
.81132 

°9 
69 

69 
6r» 

.83892 
.83966 
.84039 

74 
74 
73 

.85044 
.85116 
.85176 

66 
66 

6e 

.88237 
.  88308 
.88378 

76 

42 
43 
44 

10 

20 

30 
40 

ii.  i 

22.3 
33-5 
44-6 

II.  0 
22.0 

33-0 
44.0 

lo.o 
21.5 
32.5 
43-3 

45 
46 

8.81201 
.81270 

09 
69 

8.84113 
.84187 

74 

73 

8.85242 
.85308 

°5 

66 

68 

8.88449 
•88526 

71 

45 
46 

55-8 

55-° 

54.1 

47 

.81339 

6~ 

.84261 

74 

.85373 

°5 
68 

.88591 

7° 

47 

48 

.8140? 

60 

.84334 

73 

.85439 

°5 

66 

.88661 

48 

49 

.81475 

09 
65 

.  84408 

73 

.85505 

/r  a 

.88732 

71 

49 

50 

52 
53 

8.81545 

!  81682 
.81751 

8 
69 

68 
68 

66 

8.84481 

.84555 
.  84623 
.  84702 

73 
73 
73 
73 

8.85576 
.85626 
.85701 
.85765 

65 

65 

65 

65 

69 

8.88803 
.88873 

.88944 
.89014 

70 
76 
76 
76 

50 

52 
S3 

6    o.o 
7    o.o 
8    o.o 
9    01 

10      O.I 

54 

.81819 

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63 

.84775 

73 

•85832 

5 

/r  r 

.89085 

7° 

54 

2O      O.I 

55 

8.81888 

°8 
66 

8.84843 

73 

8.85897 

65 

69 

8.89155 

70 

55 

40    0.3 

56 

.81955 
.82025 
.82093 

°8 
68 
68 
66 

.84922 
.84995 
.85063 

73 
73 
73 

.85962 
.8602? 
.  86092 

5 

65 
65 

68 

.89225 

.89295 
.89366 

70 
76 

56 

P 

50    0.4 

59 

.82161 

°8 

68 

.85141 

73 
7-3 

.86158 

°5 

6s 

.89436 

7° 

7O 

59 

60 

8.82229 

8.85214: 

/  o 

8.86223 

*o 

8.89506 

60 

1 

Los;.  Vers 

Log.  Kxsec 

D 

Lou.  Vers. 

Log.  Exsec 

' 

P.  P. 

404 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

22°  23° 


/ 

Log.  Vers. 

Loir.  Exsec. 

D 

Log.  Vers. 

.ni:.  Exsec. 

> 

P. 

P. 

0 

8.86223 

6~ 

8.89506 

„ 

8.90034 

8.9363! 

f.C 

0 

I 

2 

.8628? 
.86352 

65 

6- 

•89576 
.89645 

7° 
70 

.90096 
.90158 

62 

6^ 

•93699 
.9376g 

07 

6? 

6*7 

i 

2 

3 
4 

.8641? 
.86482 

°5 
65 

.89715 
.89786 

69 

.90220 
.00282 

62 

•93833 
.93901 

07 
6? 

3 
4 

70 

6Q 

68 

6 

8 
9 

8.86547 
.86612 

.86675 
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.86805 

65 

64 
64 
64 

A? 

8.89856 
.89926 
.89995 
.90065 
•90135 

70 
7o 

70 
69 

8.90344 
.90406 

.90467 
.90529 
.90591 

62 

6T 
62 
6T 

6? 

8.93968 

.94035 
.94102 

.94170 
.94237 

67 
67 

6? 

67 
fi— 

I 

8 
9 

6 
7 
S 
9 

10 

•0 
3° 

J:! 

9-3 
23-3 

6-9 
8.0 
9.2 
10.3 
11.5 
23.0 
34-5 

6.8 

7-9 
9.0 

1O.2 

"•1 

22-6 

34-o 

10 

8.86870 

04 
61 

8  .  90205 

70 

8.90652 

OI 

8.94304 

07 
6? 

10 

4° 

SO 

46.£ 
58.3 

46.0 
57-5 

83 

ii 

.86934 

.90274 

6 

.00714 

6? 

•94371 

6-7 

ii 

1  2 
13 

.86999 
.87063 

64 

6/1 

.00344 
•90413 

69 

.90776 
.9083? 

6T 
61 

.94438 
.94505 

w 
67 
6-7 

12 
13 

14 

.8712? 

04 

.90483 

J 

.90899 

6T 

•94572 

07 

14 

11 

17 

18 

8.87192 
.87256 
.87326 
.87384 

64 

64 

64 

6/1 

8-90552 
.90622 
.90691 
.90766 

69 
69 
69 

8.90966 
.91021 
.91083 
791144 

61 
6! 
61 

6r 

8.94638 
.94705 
.94772 
.94839 

67 

66 

67 
62 

15 

16 

17 
18 

6 

67 

6.7 
7.8 
8.9 

66 

6.6 

7-7 
8.8 

65 

6-5 

7.6 

8.8 

19 

.87448 

04 

f\A 

.  90830 

9 

.91205 

6T 

.  94905 

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f\s 

19 

9 

10 

1O.O 

n.  i 

9-9 
it  .0 

9-7 
10.  5 

20 

21 
22 

8.87512 

•8757S 
.87646 

04 
64 
64 

8.90899 
.90968 
.91037 

69 

69 
63 

8.91267 
.91328 
•91389 

61 
61 
6r 

8.94972 

•95039 
.95105 

°6 
67 
66 

20 

21 

22 

20 

3  ' 
40 
50 

22.3 
33-5 
44-6 
55-8 

22.  O 

33-o 
44.0 
55-o 

21-6 
32-5 
43-3 
54-1 

23 

24 

.87704 
.87768 

63 
f\A 

.91105 
.91175 

09 

69 

6r» 

.91450 
.91511 

61 

6r 

.95172 
•95238 

66 

23 
24 

3 

27 
28 
29 

8.87832 
.87895 

.87959 
.88023 
.88085 

O4 

63 
64 
63 
65 

8.91244 

.91313 
.91382 

.91451 
.91520 

69 
69 

68 

8.91572 

•91633 
.91694 

•91755 
.9l8l5 

61 
61 

61 
66 

6r 

8.95305 
•95371 

•95437 
•95504 
.95576 

66 
66 

66 
65 

fit: 

25 
26 

27 
28 
29 

6 
7 

64 

6.4 
7-4 

633 

7'3 

62 

6.2 

7-2 

30 

8.88150 

61 

8.91588 

60 

8.91876 

66 

8.95635 

DO 
65 

30 

s 
9 

l'.6 

8.4 
9-4 

8.2 

9-3 

31 

.88213 

63 

•9165? 

65 

•91937 

60 

•95703 

66 

31 

to 

10.6 

10.5 

32 
33 

.88277 
.  88346 

°3 

63 
55 

.91726 
.91794 

°8 

68 
65 

.91997 
•92058 

61 

66 

•95769 
.95835 

66 

66 

32 
33 

M 

30 

40 

21.3 

32.0 
42-6 

21  .O 
42.0 

20.  6 
31.0 
4i-3 

34 

.  88404 

°3 

.91863 

°8 

£*r\ 

.92119 

•95901 

34 

5° 

53-3 

52-5 

5i.6 

35 
36 

8  .  88467 
.88536 

63 

8.91932 
.92006 

69 

68 

6X 

8.92179 
.92240 

66 

6n 

8.95967 
.96033 

66 
66 

35 
36 

39 

•88S93 
.88655 
.88720 

63 

.  92063 
.92137 
.92205 

68 
68 

.92300 
.92361 
.92421 

66 
60 

.96099 
.96165 
.96231 

66 
66 

fs 

39 

61 

60 

59 

40 

4i 

8.88783 
.88846 

63 

8.92274 
•92342 

68 

66 

8.9248? 
.92542 

66 

6r» 

8.96297 
.96362 

65 

66 

40 

6 
7 

S 

6 

I 

6.0 
7.0 
8.0 

5-9 

6.9 

42 
43 
44 

.  88909 
.88971 
.89034 

63 

.92410 
•92478 
•92546 

°8 
68 
68 

66 

.  92002 
.92662 
.92722 

60 

66 

6r» 

.96428 
.96494 
.96560 

65 

66 

68 

42 
43 
44 

9 

i  j 

2<J 

9 

10 

20.3 
30-5 

9.0 

10.  0 

20.  o 

30.0 

9-1 
19-6 
29.5 

45 
46 

8.89097 
.89160 

62 
63 

8.92615 
.92683 

°8 
68 
68 

8.92782 
.92842 

60 
60 

8.96625 
.96691 

°5 
65 
66 

45 
46 

9* 

*  9 
5°-8 

50.0 

39-3 
49.1 

47 

•89223 

63 

.92751 

68 

.92002 

60 

.96757 

5? 

47 

48 
49 

.89285 
.89348 

62 

S  - 

.92819 
.92887 

68 

SQ 

.92962 
.93022 

60 
f^. 

.96822 
.96888 

°5 

65 

48 
49 

50 

8.89411 

63 

63 

8.92955 

DO 

8  .  93082 

OO 

8.96953 

6c 

50 

6 

51 

.89473 

62 

.93022 

68 

.93H2 

59 
60 

•97018 

u:> 

51 

6 
7 

0.0 

o.o 

52 
53 

54 

.89536 

•89598 
.89666 

62 
62 

62 

.93090 
•93158 
.93226 

69 

68 
/•£ 

.93202 
.93261 
•93321 

59 

60 

r  A 

.97084 

•97149 
.97214 

65 

65 

6r 

52 
53 

54 

8 
9 

10 
20 

o.o 

O.I 
O.I 
O.I 

56 

8.89723 
.89785 

2 
62 

8.93293 
.93361 

67 
68 

6? 

8.93381 
.93446 

59 
59 
60 

8.97280 
•97345 

65 
65 

68 

55 
56 

3° 
40 
50 

O.2 
0.4 

5^ 
I  59 

.89847 
.89910 
.89972 

62 
62 
62 

.93429 
•93496 
.93564 

67 
6? 
6? 

.93500 
9356o 
.93619 

59 
59 

.97410 

•97475 
•97540 

65 
65 

6? 

57 
58 
59 

00 

8.90034 

8.93631 

8.93679 

59 

8  .  97606 

00 

' 

Lop.  Vers. 

7> 

l.'i-'.  Exsec. 

7> 

line.  Vers. 

*> 

jO«.  Exser. 

J> 

' 

P. 

V. 

405 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

24°  25° 


; 

Los.  Vers. 

z> 

Log.  Exsee 

j> 

Loar.  Vers. 

n 

Log.  Kxsec. 

J> 

' 

P. 

P. 

0 

8.93679 

8  .  97606 

, 

8.97176 

9.01443 

63 

0 

2 

.93738 

•9379? 

59 
59 

.97671 
.97736 

65 

.9722? 
.97284 

57 

58 

.01505 
.01563 

63 

I 

2 

3 

4 

.93857 
.93916 

59 
59 

rA 

.97801 
•97865 

64 

•97341 
•97398 

57 
57 

.01631 
.01694 

63 

3 
4 

65 

64 

63 

6 

7 

8-93975 
•94034 
.94094 

59 
59 
59 

8.97936 

•97995 
.  98060 

65 

64 

8-97455 
•975II 
•97568 

57 
58 

si 

9.01758 
.Ol8l§ 
.01882 

63 
62 
62 

6 
7 

6 

7 
8 
9 

6-5 
7.6 

9-7 

6.4 

1:1 

9.6 

6.5 

I: 

9-4 

8 
9 

.94153 
.94212 

59 

.98125 
.98190 

65 

67 

.97625 
.97681 

50 

58 

C2 

.01944 
.  0200? 

63 

6  a 

8 
9 

10 
20 
JO 

10.3 

21.  g 

32-5 

I0'6 
21.3 

32.0 

10.5 

21  .O 

S'-S 

10 

1  1 

12 

8.94271 
•94330 
.94389 

59 
59 
59 

8.98254 
•98319 
.98383 

04 
64 

64 

8.97738 

•97795 
.9785? 

56 
57 
58 

9.02070 
.02132 
.02195 

62 
62 

10 

ii 

12 

40 

50 

43-3 
54-1 

42.5 

53  3 

42.0 
52-5 

13 

•  94448 

59 

CO 

.98448 

61 

.97908 

*§ 

.02257 

63 

13 

H 

•945°6 

58 

.98513 

04 

•97964 

r6 

•02319 

63 

14 

15 

16 

17 
18 

19 

8.94565 
.94624 
.94683 
.94742 
.94806 

59 
59 
58 
59 
58 

8.98577 
.98642 
.98705 
•98776 
.98835 

64 
64 

64 

64 

8.98026 
.98077 
•98133 
.  98  i  90 
.98246 

56 
58 
58 
58 

56 

9.02382 
.02444 
.02506 
.02569 
.02631 

62 
62 
62 
62 

17 

18 
19 

{ 

9 

10 

62 

6.2 

9-3 
10.3 

61 

6. 

7- 
8. 
9- 
10. 

60 

6.0 
7.0 
8.0 

JO  O 

20 

21 

22 

.94917 
.94976 

58 
58 
58 

CO 

8.98899 

•98963 
.99028 

64 

64 

8.98302 
.98358 
.98414 

58 

56 
56 

c6 
™ 

9.02693 
.02755 
.02817 

62 
62 
63 

20 

21 

22 

K 

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40 

20.  6 

31.0 
41-3 
5i-6 

20.3 
3°  -5 
40.5 
50-8 

20  o 
30  o 
40  o 
50.0 

23 

•95034 

58 

.99092 

6/T 

.98476 

.02880 

23 

24 

53 

•99156 

04 
6  i 

.98527 

56 

.02942 

60 

24 

25 

8.95151 

8 
53 

8.99226 

04 
61 

8.98583 

6 

9.03004 

25 

26 

.95210 

8 

rQ 

.99284 

04 
6/1 

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c6 

.  03066 

£2 

26 

27 
28 
29 

.95268 
•95326 
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5° 
58 
58 

ro 

•99348 
.99412 

•99476 

04 
64 
64 

6/1 

.98695 
.98756 
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5° 

55 

56 

r6 

.03128 
.03190 
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62 
62 
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27 
28 
29 

6 
7 

8 

59 

5-9 
6.9 

7-§ 

58 

5-8 
6.7 

7-7 

57 

5-7 
7-6 

30 

32 
33 

34 

8-95443 
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•95559 

.95675 

58 
58 

5? 
58 
58 

co 

8.99546 
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.99663 
.99732 
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04 
64 
64 
64 
63 
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8.98802 
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.99030 
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56 
56 

55 

56 

55 

9-033I3 

•03375 
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-.03499 
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OI 
62 
62 

6T 
62 

6T 

30 

32 

33 
34 

9 

10 

20 

3° 

40 

50 

8.  8 
9-8 
19-6 
29-5 
39-3 
49-1 

8.7 
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19-3 
29.0 

48-1 

8-5 
9-5 
19.0 
28.5 
38.0 
47-5 

35 
36 

8-95733 
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58 

58 

8.99860 
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04 

63 
6/1 

8.99141 
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56 

cp 

9.03622 
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61 

67 

35 
36 

37 

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8.9998? 

U4 
65 

.99252 

55 

r? 

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6T 

37 

39 

.95907 
•95965 

58 

9.00051 

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uj 
63 
6/1 

.99308 
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55 

55 

5P 

.03807 
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6f 

6? 

38 
39 

6 

56 

5-6 

55 

5.5 

54 

5.4 

40 

41 
42 

43 
44 

8.96023 
.  96086 
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.96196 
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9 

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57 

ra 

9.00173 
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04 
63 
63 
63 
63 
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8.99419 
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5 
55 
55 
55 
55 

9.03930 
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OI 

6T 
6T 
6T 
6! 

6T 

40 

42 
43 
44 

7 
S 
9 

10 
20 
90 

40 

H 

8.4 
9-3 

28.0 

S'l 

6.4 

5:1 

9.1 
i8.§ 
27-5 
36.6 

6.3 

7.2 

8.1 
9-o 
18.0 

5>7.0 

36.0 

45 

8.96311 

57 

9.00495 

61 

8.99695 

55 

rP 

9.04238 

01 

AT 

45 

JU 

46.6 

4S«  8 

45-o 

46 

.96363 

8 

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63 

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55 

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6T 

46 

47 
48 

49 

.96426 
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It 

Sf 

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.00622 
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63 
63 

.99806 
.99861 
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55 

55 
55 

.04366 
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61 
6! 

47 
48 
49 

50 

8.96593 
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57 
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9  .  008  i  2 

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63 
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8,99971 
9.00025 

55 
55 

9.04544 
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6T 
5T 

50 

6 
7 

0.6 
0.6 

52 

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5 

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63 

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55 

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61 

52 

8 

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53 

54 

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57 

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53 

63 

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55 
55 

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61 

6? 

53 

54 

9 

10 

20 

O.I 
O.I 

55 

8.96885 

57 

9.01128 

6 

9.00245 

55 

9.04850 

6r 

55 

3° 
40 

o.§ 

56 

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57 

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6 

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55 

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6r 

56 

50 

0.4 

57 
58 

.96999 
.97058 

57 

.01254 
.01317 

63 

67 

•00355 
.0041  i 

54 

.04972 
•05033 

6l 
6r> 

57 
58 

59 

.97H3 

57 

.01380 

53 
6l 

.00466 

55 

CA. 

•05093 

6l 

59 

60 

8.97176 

9.01443 

9.00520 

9.05154 

60 

' 

Log.  Vers. 

i  J> 

Log.  Uxse<*. 

D 

Loe.  Vers. 

Lour.  Kxsec. 

' 

P. 

P. 

406 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

26°  27° 


/ 

Log.  Vers. 

D 

Log.  Exsec.  1> 

Log.  Vers.  1> 

Log.  Exsec.   D 

• 

P.  P. 

0 

I 

2 

9.00520 

.00575 
.00630 

55 

1  54 

9.05154 
.05215 
.05276 

61 
61 

fin 

9.03740 
.03792 
'  .03845 

52 
52' 

9.08752 
.O88ll 
.08870 

59 
59 

0 

2 

3 

.00684 

;  54 

•05337 

fir 

.03898 

?? 

.08929 

59 

3 

4 

.00739 

54 

.05398 

fin 

.03950 

52 

.08988 

59 

4 

ftt    f\r\    Cr\ 

i 

7 

9.00794 
.00843 
.00903 

55 
54 

54 

9-05458 
.05519 
.05580 

61 
66 

fin 

9  .  04002 
.04055 
.04107 

52 
52 

ll 

9.09047 
.09106 
.09164 

59 
59 
58 

1 

7 

6 
7 
8 
9 

jy 

6.1    6.0    5.9 
7.1    7.0    6.9 
8.1    8.0    7.8 
9.1    9.0    8.{j 

8 

.00957 

54 

.05646 

fin 

.04160 

§ 

.09223 

59 

8 

10 

10.  i   10.  o    9.3 

9 

.01011 

54 

.05701 

.  042  1  2 

52 

.09282 

59 

9 

20 

3° 

20.3     20.0     19.  £ 

30.5   30.0   29.5 

10 

ii 

9.01066 
.01126 

54 
54 

9.05762 
.05822 

66 
fi6 

9.04264 

.043*7 

P 

9.09341 
.  09400 

58 
59 

50 

10 

ii 

40 
50 

40.  1  40.0  39.  1 
50.8  50.0  49.1 

12 
13 

.01174 
.01229 

54 
54 

.05883 
•05943 

66 

fin 

.04369 
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2 
52 

.09458 
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O 

59 

r  6 

12 
13 

14 

.01283 

54 

r  A 

.06004 

.04473 

52 

.09576 

5o 

15 

16 
17 

9.01337 

.01391 

.01445 

54 
54 
54 

9.06064 
.06124 
.06185 

60 

66 

9.04525 

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.  04630 

52 
52 
52 

9.09634 
.09693 
-09752 

58 
58 
5? 

II 

17 

6 
7 

58   57 

i?    I'* 

6.7     6.5 

18 

.01499 

54 

.06245 

fin 

.04682 

52 

.09816 

58 

18 

8 

19 

.OICCJ4. 

54 

.06305 

AA. 

•04734 

52 

.09869 

58 

19 

9 
10 

9-<j     9-5 

20 

21 

^7  "> 

9.0l6o3 
.01662 
.01715 

54 

11 

9.06366 
.06426 
.06485 

OO 
60 

66 

fin 

9.04786 
.04837 
.04889 

52 
5? 
52 

9.09927 
.09986 
.  10044 

58 
58 

58 
C3 

20 

2  1 
22 

20 

3° 
40 
50 

19.3    19.0 
29.0    28.5 
38.  §    38.0 
48.3    47-5 

23 

.OI76§ 

54 

.06545 

fin 

.04941 

5- 

.10102 

3° 

23 

24 

.01823 

54 

.06605 

/T5 

-04993 

52 

-y 

.  I0l6l 

58 

ro 

24 

3 

9.01877 
.01931 

54 

53 

9.06667 
.06727 

OO 
60 
fin 

9.05045 
.05097 

51 

52 

r  t 

9.I02I§ 
.  10278 

58 
58 

r« 

25 
26 

27 

.01985 

54 

.06787 

fin 

•05148  *„ 

.10336 

5° 

27 

55    54 

28 

.02038 

53 

.06847 

fin 

,05200 

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.10394 

58 

28 

6 

5-5     5-4 

29 

.02092 

54 

.06907 

.05252 

2 

.  10452 

CO 

29 

•1 

8 

7-3     7-2 

30 

32 

9.02146 
.02199 
.02253 

53 
53 
54 

9.06967 
.07027 
.07087 

60 
60 
en 

9-05303 
.05355 
.05407 

52 

9.  I05II 
.  10569 
.  10627 

58 
58 
58 

30 

32 

9 
10 

20 
30 

8.2     8.1 
9.1     9.0 
18.3    18.0 
27.5    27.0 

33 

.02307 

53 

55 

.07145 

59 

fin 

•05458 

5f 

.10685 

58 
cS 

33 

40 

36.^    36.0 

34 

.02366 

3 

.07205 

.05510 

r  c 

•  10743 

5° 

ro 

34 

35 

9.024U 

53 

9.07265 

9.05561 

51 

9.  I080I 

5° 

rQ 

35 

36 

37 

.02467 
.02521 

53 
53 

.07326 
.07386 

59 

60 

.05613 
.05664 

51 
5i 

.10859 
.1091? 

5° 
58 

c8 

36 
37 

38 
39 

.02574 
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53 
53 

•07445 
.07505 

60 

r  A 

.05715 
.05767 

5* 

_« 

.10975 
•II033 

58 

38 
39 

53   52 

6     5.3      5-2 

40 

9.O268I 
.02734 

53 
53 

c? 

9.07565 
.07624 

59 
59 

9-05818  i: 

.05869  5{ 

9.  II09I 
.11149 

$ 

40 

7 
8 
9 

6.2     6.6 
7.6     6.9 
7.9     7.8 

42 
43 
44 

.0278? 
.02846 
.02894 

53 

53 
53 

.07684 

.07743 
.07803 

59 
59 
60 

.05921 
.05972 
.06023 

11 

.  1  1  207 
.11265 
.11323 

57 
58 
58 

p  A 

42 
43 
44 

10 

20 

3° 

40 

8.§     8.g 
J7-6    J7-3 
26.5    26.0 

35-3    34,-6 

45 

9.02947 

53 

9.07863 

9.06074 

5" 

9.  1  1386 

5/ 

45 

50    44.1    43.3 

46 

.03000 

:>3 

.07922 

59 

.06125 

51 

«  !  J438 

cc 

46 

47 

.03053 

53 

.07981 

59 

to. 

.06175 

51 

.11496 

5° 

rfj 

47 

48 
49 

.O3I06 
.03159 

53 

53 

.08041 
.08106 

59 
59 

.0622? 
.06279 

5i 

•"55* 
.11611 

57 

57 

CQ 

48 
49 

50 

52 

9.03212 
.03265 
.03318 

53 
53 
53 

9.08160 
.08219 
-08273 

59 
59 
59 

rA 

9.06330 
.06386 
.06431 

51 
50 
5i 

9.11  669 
.11727 
.11784 

55 

1 

50 

52 

51    o 

6    5.1    0.6 

7    5-9    o.o 
8    6.8    o.o 

53 

.03371 

53 

.08338 

59 

.06482 

51 

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pfj 

53 

9    7-6    o-1 
10    8.5    o.i 

54 

52 

.08397 

59 

rA 

.06533 

51 

.11899 

57 

54 

20     17.0     O.I 

55 

9-°3476 

53 

9.08455 

59 

9.06584 

si 

9.U95? 

5. 

55 

40     34.0      0.§ 

56 

57 
58 
59 

-03529 
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.03634 
•0368? 

53 

i 

53 

c5 

.08515 
.08574 
.08634 
.08693 

59 
59 
59 
59 

CQ 

.06635 
.06686 

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)  K.  <Q  N-  <C 

r,  ID  \f\  \i*>  u 

.  12015 
.12072 
.12129 
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% 

57 

56 

I 

59 

50     42.5      0.4 

60 

9.03740 

9.08752 

9.06838 

9.12244 

60 

Log.  Vers.!  J) 

Lour.  Exsec.   7> 

Log.  Vers.   />   Log.  EXKCC.'  D 

P.  P. 

407 


TABLE   VIII. —LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

28°  29 


/ 

Log.  Vers. 

MX.  Exsec. 

D 

Log.  Vers. 

D 

josr.  Exsec. 

D 

/ 

P. 

P. 

0 

I 

2 

3 

4 

9.06838 
.06883 
•06939 
.  06996 
.  07046 

53 
51 
50 
50 

rr\ 

9.12244 
.12302 
.12359 

.12416 
.12474 

1 

9.09823 
.09872 
.09926 
.09969 
.  IOOI8 

49 
48 
49 
48 

9.15641 
.15697 
.15752 
.15808 
.15864 

56 

55 

1 

0 

i 

2 

3 

4 

6 

7 

5? 

5-7 
6-7 

57 

:I 

fg 

6 

8 
9 

9.07091 
.07141 
.07192 

.07242 
.07293 

5° 
50 

9.12531 
.12588 

.12703 
.  12760 

1 

57 
5? 
57 

9.  10067 
.10115 
.  10164 
.  I02I3 
.  IO26T 

49 
48 
48 
49 

48 
.5 

9.15920 

•15975 
.  16031 
.16087 
.  16142 

56 

6 

8 
9 

8 
Q 

10 
20 

je 

40 

5° 

7-6 
8.6 
9.6 
19.1 
28.7 
38.3 
47-9 

7-6 
8-5 
9-5 
19.0 

38.0 
47-5 

7-5 
8-5 
9-4 

iS.j 

28.2 

37-6 
47.1 

10 

ii 

12 
13 

9-°7343 
•97393 
.07444 
.07494 
.07544 

5° 

5§ 

50 
5o 

9.  12817 
.12874 
.12931 
.12983 
.13045 

57 

9 

57 
57 

9.  10310 

•10358 
.  10407 
.10455 
.  10504 

48 
48 
48 
48 
48 

9.16198 
.16254 
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.16365 
.  16426 

55 

55 
55 

c? 

10 

ii 

12 
13 
H 

7 

56 

5-6 

55 

I:! 

55 

e 

15 

16 

17 
18 

19 

9.07594 
.07644 
.07695 
.07745 
-0779s 

5° 
50 
50 
50 
50 

9.13102 
•I3I59 
.13216 

.13273 
.133^0 

57 
57 
57 
56 
57 

9.10552 
.  10601 
.  10649 
.  1069? 
.  10746 

48 
48 
48 

48 

48 

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9.16476 
.16531 
.16587 
.16642 
.16698 

55 
55 
55 
55 
55 

15 

16 

17 
18 

19 

9 

10 

20 
30 

40 

5° 

9-3 
18.  g 
28.0 
37-3 
46-  6 

8.3 

*i 

27.7 

ll'.s 

9.1 
18.3 
27.5 
36.6 

45-8 

20 

21 
22 
23 
24 

9.07845 
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•07945 
•07995 
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5o 
5o 

9-133*7 
.13444 
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.1355? 
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57 

57 
56 

9.10794 
.  10842 
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•  10939 
.  10987 

45 

48 
48 

48 
48 

.  0 

9.16753 
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.  16919 
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55 

55 
55 

20 

21 

22 
23 
24 

6 
7 
8 

54 

7.2 

54 

K 

7.2 

25 
26 
27 
28 
29 

9.08095 
.08145 
.08195 
.08244 
.08294 

5° 

5o 
49 
50 

9.13671 
.1372? 
.13784 
.13841 
.1389? 

57 
56 

9.11035 
•  11083 
.11131 
.11179 

.  1122? 

45 

48 

48 

48 
48 

.0 

9.17029 
.17085 
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-  .17195 
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55 

55 
55 
55 
55 

25 
.26 

27 
28 

29 

9 

10 
20 
30 

4° 

50 

8.2 

9.1 
18.1 
27.2 

36.3 
45-4 

8.1 
9.0 
18.0 
27.0 
36.0 
45.0 

30 

32 
33 

34 

9.08344 
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•08493 
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49 
50 
49 
49 
50 

A  f\ 

9-13954 
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.1406? 
.14124 
.  14186 

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56 
56 

56 

r  ~ 

9.11275 
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.  i  1467 

45 

48 
48 

49 

48 

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9.17305 
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55 

1 

55 

30 

32 
33 
34 

6 
7 
S 

51 

5-9 
6.8 

56 

5-9 
6.7 

50 

5-0 

11 

35 
36 

P 

39 

9.08592 
.  08642 
.08691 

.08741 
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49 
49 
49 
49 
49 

9-I42j7 
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56 
56 
56 

56 

9.11515 
.  11562 
.  11616 
.11653 
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48 

49 

48 
48 

49 

9.17581 
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.17691 
.17746 
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55 
55 
55 

55 
55 

35 
36 
37 
38 
39 

y 

1C 

20 

S" 

40 

5° 

7-6 
8-5 
17.0 
25.5 

42.5 

7.6 
8.4 
x6.5 
25.2 
33-6 
42.1 

5: 

16  £ 
25.0 

33-3 
41-6 

40 

42 
43 
44 

9.08840 
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.08939 
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.090^7 

49 
49 
49 
49 
49 

A  A 

9.14519 

•'457§ 
.  14631 
.  14688 
•  H744 

5b 
56 
56 
56 
56 

9.11754 
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48 
4? 
4? 
48 

4? 

f- 

9.17856 
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.17965 
.  18026 
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55 
54 
55 
55 
54 

40 

42 
43 
44 

6 

49 

4.9 

fl 

7-4 

49 

4-9 

5-7 

7-3 

48 

4.8 

ii 

45 
46 

47 
48 

49 

9.09087 
.09136 
.09185 
.09234: 
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49 
49 
49 
49 
49 

9.  14806 
.14855 

.14913 
.  14969 
.15025 

56 
56 
56 
56 
56 

9.11992 
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47 
49 
4f 
4? 
49 

A  £ 

9-  18130 
.18185 
.18239 
.18294 
.18349 

55 
55 
54 

11 

II 

47 
48 

49 

10 
20 

3° 
40 

50 

8.2 

16.5 
24.7 
33-o 
41.2 

8-1 
16.3 
24.5 
32-6 
40-8 

8.1 
16.1 
24.2 

40.4 

50 

52 
53 
54 

9-09333 
.09382 
.09431 
.09486 
•09529 

49 
49 
49 
49 
49 

9.15081 

.15193 
.15249 

•15305 

^ 
56 

56 
56 

9.  12229 
.12277 
.12324 

.12371 
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47 
4? 
47 
49 

4? 

9  •  i  8403 
•18458 
.18513 
•1856? 
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54 

y 

54 
54 

50 

52 

53 
54 

r, 
7 

B 
9 

48 

4.8 
5-6 
6.4 
7.2 

4? 

4-7 

M 

47 

4-7 
7.6 

55 

56 

11 

59 

9-09578 
.0962? 
.09675 
.09725 
•09774 

49 
49 
49 
48 
49 

9.15361 
•I54if 

.15473 
.15529 
.15585 

^ 
56 

56 

y 

9  .  i  2466 
.12513 
.12566 
.12608 
.12655 

47 

49 

47 
4? 
47 

A.7 

9.18675 
.18731 
.18786 
.18846 
.18894 

54 
54 

11 

54 

55 
56 

59 

20 

40 
JO 

16.0 
24.0 
32.0 
40.0 

7-? 
IS-  § 
23-7 
3T-6 
39-6 

15-6 
T5 

60 

9.09823 

49 

9.15641 

->u 

9.  12702 

9  •  i  8949 

00 

Loe.  Vers. 

7> 

Loe.  Exsec 

/> 

Lot.'.  Vers. 

!  J> 

Log.  Exsec. 

' 

P 

P. 

TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

3O°  31° 


Lo*.  Vers. 

2> 

Log.  Kxsec. 

Z> 

Log.  Vers. 

It 

Los.  Kxsec. 

J> 

P 

P. 

0 

I 

2 

3 

4 

9.  12702 

.12749 
.12796 
.12843 
.12896 

47 
47 
47 
47 

9.18949 
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.19058 
.19112 
.19167 

;  54 

54 
54 
54 

9-15483 
,  .15528 
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.15665 

45 

45 
45 

9.22176 
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.22282 
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•22383 

53 
53 
53 

0 

i 

2 

3 

4 

cl 

ro 

6 

8 
9 

9.12937 
.12984 
.13031 
.13078 
.13125 

47 
47 
47 
47 
47 

9.I922I 

.19275 
.19329 

.19384 
.19438 

It 

l\ 
54 

r  A 

9.I57IO 

.15755 
.15801 

.15846 
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45 
45 
45 
45 

45 

9.22441 
•  22494 
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.22653 

53 
53 
53 
53 
53 

6 

8 
9 

t 

7 
- 

9 

i  , 
1  1 

-  -^ 

54 

11 

7.2 

8.2 

,1:1 

27.2 

5-4 
6-3 
7.2 
8.1 
9.0 
18  o 
27.0 

53 

1.1 
ll 

8-9 

3.1 

10 

ii 

12 
13 

14 

.13219 
.13266 

9.19492 

•19546 
.19601 

.19655 
.19709 

54 
54 
54 
54 
54 

9-15937 
.15982 
.16027 
.16073 

45 
45 
45 
45 
45 

9.22705 
.22759 
.22812 
.22865 
.22918 

53 
53 
53 
53 
52 

10 

ii 

12 
13 
14 

«o 

50 

36.3 

45-4 

36.0 

35-6 
44.6 

15 

16 

17 
18 

19 

9.13406 
.13453 
.13500 
.13546 
.13593 

47 
46 

9.19763 
.19817 
.19871 
.19925 
.  I997Q 

54 
54 
54 
54 

9.16163 
.16203 
.16253 
.16293 
•16343 

45 
45 
45 

45 
45 

9.22971 
.23024 

•23076 
.23129 

.23182 

53 
53 
52 
53 
52 

15 

16 

17 
18 

19 

6 

7 

9 

10 

53 

7'° 
7-9 
8-8 

52 

1:1 

7.0 

7-9 
8.7 

52 

5-2 
6.6 
6.9 
7.8 

20 

21 

22 
23 
24 

9-  i  3639 
.13685 
.13733 
.13779 
.13826 

i 

9.20033 
.20087 
.20141 
.20195 

.  20249 

54 
54 
54 
54 
54 

9.16383 
.16434 

.16479 
.16523 
.16563 

45 
45 
45 
44 
45 

.23287 
.23340 

.23393 
.23446 

53 
52 

Ii 

53 

20 

21 

22 
23 
24 

23 

30 
40 

3:f 

35-2 
44-1 

26^2 
35-0 
43-7 

34-6 
43-3 

26 

27 
28 
29 

9.13872 
.139^9 
.13965! 
.  1401  i 

.  14058 

A? 

9.20303 
•20357 
.20411 
.  20465 
.20513 

53 
54 
54 

n 

9.16613 
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.16703 
•16743 
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4b 
45 
45 
45 
44 

9-23498 
.23551 
.23603 
.23656 
.23709 

52 

1 

53 

-a 

27 
28 
29 

6 

| 

47 

4-7 

46 

4-6 
5-4 

6.2 

30 

32 

33 
34 

9.14104 
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46 

9.20572 
.20626 
.20680 
•20733 
.20787 

54 
53 
54 
53 
54 

c5 

9.16838 
.16882 
.16927 
.16972 
.17017 

45 
44 
45 
44 

45 

.  j 

9.23761 
.23814 
.23865 
.23919 
.23971 

52 

52 
52 

30 

32 
33 
34 

9 

10 

j  ! 
SO 

JO 

I" 

i7'5 
23.7 
3T-6 
39.6 

7.0 

isl 

31-3 
39-1 

23.2 
31.0 
38.7 

35 
36 

39 

9-I433t> 

.{. 

9.20841 
.20894 
.20943 

.21002 
.210=^ 

53 
53 

52 

9.17061 
.  17106 

.17195 
.17240 

44 
44 
45 
44 
44 

9.24024 
.  24076 
.24123 
.24181 

.24233 

53 

52 
52 
52 

52 

35 
36 
37 
38 
39 

6 

46 

4.6 

1S, 

45 

4.5 

40 

41 

42 

43 

44 

9.H5% 
.14612 
.14653 

.14704 

.I47;6 

40 
46 

9.21109 
.21162 
.21216 
.2I26§ 
.21323 

53 
53 

i 

9.17284 
.17329 

.17373 
.17418 
.17462 

44 
44 
44 
44 

44 

9.24285 

.24338 
.24396 

.24442 
-24495 

52 

52 

52 

40 

42 
43 
44 

7 
- 

) 

10 

M 
30 

1:1 

6.9 

*5-3 
23.0 

1:1 

6.8 
7-6 

22.7 
30.3 

6'.° 
7-5 
15-0 
22.5 
30.0 

45 
46 
47 
48 
49 

9-14796 
.  14842 
.14883 

.  H934 
.  14980 

46 
46 
46 
45 

9-21376 
.21430 
•21483 

•21537 
.21596 

1 

53 
53 

9.17507 

.17551 
.17596 
.  I  7640 
.17684 

44 
44 
44 
44 
44 

9.24547 
.24599 
.24651 
.24704 
.24756 

52 

52 

P 

52 

4! 

46 

47 
48 
49 

ID 

38-3 

37-9 
i 

37-5 

50 

51 

5* 
53 

54 

9.15026 
.15071 
.1511? 

-15163 

.15209 

45 
46 
46 

45 

A  ? 

9.21643 
.21697 
.21756 
.21803 
.21857 

53 
53 
53 
53 
53 

9.17729 

•17773 
.1781? 
.17861 
.17906 

44 
44 
44 
44 
44 

9.24808 
.24860 

.24912 
.24964 
.25015 

52 

| 

52 

50 

S2 
53 
54 

.  6 
1 
E 

9 
10 

•L  \ 

4' 

A 

5 
5 
6 

7 
14 

i^   ' 

4 

t 
9 
7 

4 
1    ' 

K 

*•* 

5-i 
5-8 
6.6 

7-1 
4-6 

55 
56 
57 
58 
59 

9.15254 
.15306 
.15346 
.15391 
•15437 

45 
46 
45 
45 

9.21910 
.21963 
.22015 

.  22070 
.22123 

53 
53 
53 
53 
53 

C-3 

9.17950 

-17994 
.18033 
.18082 
.18125 

44 
44 
44 
44 
44 

A  A 

9.25063 
.25126 
.25172 
.25224 
•25276 

52 

52 
52 
52 
S2 

11 

57 
58 
59 

4« 
jo 

29 
37 

6    2 
i    3 

H 

5.  6 

60 

9.15483 

9.22176 

!    JJ 

9.  18176 

9.2532§ 

60 

' 

Log.  Vers. 

Los:.  Kxsec. 

Log.  Vers. 

I> 

josr.  Kxsec. 

i) 

7 

P. 

P. 

400 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL    SECANTS. 

32°  33° 


Log.  Vers. 

D 

Log.  Exsec. 

Log.  Vers. 

Logr.  Exsec. 

D 

/ 

P. 

P. 

0 

2 

3 

4 

9.  18176 
.18214 
.18253 
.18302 
•18345 

44 
44 
44 
44 

9-25328 
.25386 

.25432 
.25484 
.25536 

52 
52 

9.20771 
.20814 
.20855 
.  20899 
.  20942 

42 

42 

43 

9.28412 
.  .28463 
.28514 
.28564 
.28615 

51 
51 
56 
51 

0 

I 

2 

3 

4 

ro 

CT 

1 

7 
8 

9 

9.18396 
.18434 
•18478 
.18522 
.18566 

44 
44 
44 
44 
43 

9.25588 
.25640 
.25692 
.25743 
•25795 

52 
52 

cf 

9  .  20984 
.21027 
.21069 

.21112 

•2II54 

42 
42 
42 

42 
42 

9.28665 
.28717 
.28768 
.288l§ 
.  28869 

51 
5o 

6 

8 
9 

6 
7 
8 
9 
10 
20 
3° 

5Z 

6\l 
7.8 

J7-3 
26.0 

5* 

7-7 
8.6 

25!? 

5-1 

5-9 
6.8 

7-6 
8-5 
17.0 
25'5 

10 

ii 

12 

13 

H 

9.18610 
.18654 
.1869? 

.18741 
.18785 

44 
44 
43 
44 
43 

9.25847 
•25899 
.25956 
.26002 
.  26054 

51 

52 
51 

11 

r? 

9.2II96 
.21239 
.2I28T 
.21324 
.21366 

42 
42 
42 
42 

9.  28920 
.  28976 
.  2902  1 
.  29072 
.29122 

50 
50 

10 

ii 

12 
13 
H 

40 
5° 

34-!? 
43-3 

34-3 
42.9 

34-o 
42.5 

jl 

18 

9.18829 
.18872 
.18916 
.18959 
.19003 

44 
43 
43 
43 
44 

AT. 

9.26105 
.2615? 
.  26209 
.  26266 
.26312 

51 
52 
51 
51 
51 

9.21408 
.21451 
•21493 
•21535 

.2157? 

42 
42 
42 
42 
42 

9.29173 
.29223 
.29274 
.29324 
•29375 

51 
50 
5° 
50 

15 

16 

17 
18 

19 

6 

7 
8 
9 

10 

56 

5-o 

8^4 

50 

5'° 

H 

6.6 

49 

si 

6.6 

20 

21 

22 
23 
24 

9.19047 
.19096 
.19134 
.1917? 
.19221 

43 
43 
43 

43 

9.26364 

.26415 
.  26467 
.26518 
.26570 

52 

9.21620 
.21662 
.21704 

•21746 
.21788 

42 
42 
42 
42 
42 

9.29426 

•29476 
•29527 
•29577 
.2962? 

5° 
50 
55 
50 
50 

20 

21 

22 

23 
24 

20 

30 
40 
50 

16.5 
25.2 
33-6 
42.1 

25-0 
33-3 

24.7 
33-0 
41.2 

25 
26 

27 
28 
29 

9  .  i  9264 
.  i  9308 
.19351 
•19395 
.19438 

3 

43 

43 
43 

43 

A  5 

9.26621 

.26673 
.26724 
.26776 
.26827 

51 

H 

51 

r  r 

9.21836 
.21872 
.21914 
.21955 
.21998 

42 
42 
42 
42 
42 

9.29678 
.29728 
.29779 
.29829 
.29879 

5° 
50 
50 
50 
5o 

11 

27 
28 

29 

6 

i 

44 

4.| 

5'8 

4§ 

4-3 

[:S 

43 

4-3 
5.7 

30 

32 
33 
34 

9.19481 
.19525 
.19568 
.19611 
.19654 

43 
43 

a 

43 

9.26873 
.26930 
.  26981 
.27032 
.  27084 

51 
51 
51 

rT 

9.  22046 
.  22082 
.22124: 
.22165 

.2220§ 

42 
42 
42 
42 
42 

9.29930 
.  29986 
.30036 
.30081 
.30131 

5° 
50 

5° 

30 

32 
33 
34 

9 
10 
20 
30 
40 
So 

6.6 
^•3 

22.0 
29-3 
36.6 

7-2 

29.0 

36.2 

6.4 

35-8 

35 
36 

39 

9.19698 
.19741 
.19784 
.1982? 
.  19876 

43 

43 
43 
43 

AT. 

9.27135 
.27186 
.27238 
.27289 

.27346 

51 
51 
51 
51 
51 

9.22250 
.22292 
.22334 
.22376 

.2241? 

42 
41 
42 
42 
4* 

9.3018! 
.  30231 
.  30282 

•30332 
.30382 

5° 
50 
50 
5o 
.50 

35 
36 

i 

39 

6 

4.2 

42 

4-2 

4.1 

40 

42 
43 
44 

9.19914 

•19957 
.20000 
.20043 
.  20086 

43 

43 
43 

43 

9.27391 
•  27443 
.  27494 
•27545 

.27596 

5\ 
51 

\\ 

9.22459 
.22501 
•22543 
.22584 
.22626 

42 
4i 
42 

41 
41 

9-30432 
.30482 

.30533 
.30583 
-30633 

5° 
50 
50 
50 
50 

40 

42 
43 
44 

7 
8 

9 

10 

20 

3° 
40 

4-9 

5-6 
6.4 

7-i 

21.2 

5-6 
6-3 
7.0 
14.0 

21.  O 

28.0 

4.§ 

13-  § 
20.  7 

27.6 

45 
46 

47 
48 

49 

9.20129 
.20172 
.20215 
.20258 
.20301 

43 
43 
43 
43 
43 

9-2764? 
.27693 

•  27749 
.27806 
.27852 

Sl 

51 
51 

9.22668 
.22709 
.22751 
.22792 
.22834 

42 
41 
41 
41 
41 

9.30683 
•30733 
•30783 
.30833 
.30883 

5° 
50 

5o 

45 
46 
47 
48 
49 

5° 

35-4 

35-o 

34.6 

50 

52 
53 
54 

9-20343 
-20385 
.  20429 
.20472 
.20515 

42 
43 
43 

42 

9.27903 
.27954 
.  28005 
.28056 
.28107 

51 

.2291? 
.22959 
.  23006 
.  23042 

4? 
4f 

9-30933 
.30983 
.31033 
•31083 

•3"33 

5° 
50 

50 

.  c 

50 

52 
53 

54 

6 

7 
8 
9 

10 

20 

41 

4.1 

4.5 

6.] 
13-d 

1 

59 

9.20558 
.20606 
.20643 
.20686 
.20723 

43 
42 
43 

43 
42 

AT. 

9-2815? 
.28203 
.28259 
.28316 
.28361 

5° 
5i 

CO 

9.23083 
.23124 
.23166 
.2320? 
.23248 

41 

4? 
4! 

4i 
41 

9-31183 

•31233 
•31283 
•31333 
•31383 

49 
50 
50 
50 
5° 

AQ 

55 
56 

11 

59 

30 
4° 
50 

20.5 
27-3 

60 

9.20771 

9.28412 

9.23290 

9.31432 

00 

' 

Log.  Vers. 

z> 

Log.  Exsec. 

Log.  Vers. 

MS;.  Exsec.  1 

JT> 

' 

P. 

P. 

TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

34°  35° 


!    ' 

Log.  Vers. 

Log.  Exsec.  Z» 

Log.  Vers. 

.111:.  Exsec. 

D 

'  / 

P. 

P. 

0 

9.23290 

/IT 

9-3I432 

9.25731 

9-34395 

0 

I 

.23331 

41 

.31482 

5° 

.25771 

4° 

•34444 

1« 

I 

2 

3 
4 

.23372 
.234H 
.23455 

4l 

A  ? 

•31532 
.31582 
.31632 

5° 
49 
50 

A  A 

^  .25811 
.2585t 
.25891 

40 
40 
40 

.34492 
.34541 
•34596 

48 
49 
49 

2 

3 

4 

50 

o   5.0 

49   49 

4-9   4-9 

5-8    5-7 

1 

9-23496 

.23537 

41 

/IT 

9.31681 
.31731 

49 
5° 

9.25931 

.25971 

40 
40 

'.  34688 

49 
48 

6 

9  7.5 

10   8.3 

6.6    6.5 

51  5:1 

8 
9 

.23579 

.23620 

.23661 

41 

41 
41 

.31781 
.31886 

49 
50 
49 

.26011 
.26051 
.26091 

40 

39 
40 

.34737 
.34785 
.34834 

49 
48 
49 

8 
9 

20   16.6 
30   25.0 
4°   33-3 
50   41  6 

16. 
24- 
33- 
41- 

16.3 
24-5 
o   32.  ^ 

2  40.  § 

10 

ii 

9.23702 
.23743 

41 

4* 

9.31930 
.31980 

r<_ 
5° 

49 

9.26131 
.26171 

40 
40 

9-34883 
•34932 

49 

48 

AQ 

10 

ii 

12 
13 

14 

.23784 
.23825 

41 

41 
41 

.32029 
.32079 
.32129 

4? 
49 
50 

At\ 

.26210 
.26256 
.26296 

39 
40 

40 

.  34986 
•35029 
•35078 

48 
49 

48 

12 
'3 

6 

4*1 

4>  8 
11 

48 

4.8 

15 

16 

17 
18 

19 

9.23907 

•23948 
.  23989 
.  24036 
.24071 

41 
41 

4i 
4i 

9.32178 
.32228 

.32277 
.32327 
.32377 

49 
49 
49 
49 
50 

9.26330 
.26370 
.26409 
.  26449 
.26489 

39 
40 

39 
40 

39 

_~ 

9.35127 

-35I75 
•35224 
.35273 
-35321 

49 
48 
49 
48 
48 

17 

18 
19 

9 

10 

20 
30 

40 
50 

5:1 

16.1 

24.2 

3»-3 
40.4 

7-2 

8.0 
16.0 
24.0 
32.0 
40.0 

20 

21 

9.24112 
.24153 

46 
4i 

9.32426 
.32476 

tr\ 

49 
49 

9.26523 
.26563 

39 
40 

9-35370 
•35419 

48 
49 

20 

21 

22 
23 

.24194 
.24235 

4l 

41 

.32525 
•32575 

49 
49 

.26608 
.2664? 

39 
39 

.3546? 
•355^6 

48 
48 

22 
23 

6 

41 

4.1 

41 

4.1 

24 

.24275 

4° 

.32624 

49 

.26687 

39 
-~ 

.35564 

48 

24 

7 
8 

4-8 

5-5 

4.8 
5-4 

25 
26 
27 
28 
29 

9-243^6 
•24357 
.24398 
.24438 
.24479 

41 

46 

40 

9.32673 

.32723 
.32772 
.32822 
.3287I  ! 

49 
49 
49 
49 
49 

9.26726 
.26765 
.26806 
.26845 
.26885 

39 

40 
39 
39 
39 

tf\ 

9-356I3 
.35661 
•35710 
•35758 
-35807 

48 
48 
48 
48 
48 

25 
26 

27 
28 
29 

9 
10 
20 
3° 
40 
5° 

6.2 

6.9 
I3.| 

20.7 

27-6 
34-6 

6.1 

6.5 
13-6 

20.5 

2J1 

30 

9.24520 

4° 

9.32926 

49 

9.26924 

39 

9.35855 

48 

AQ 

30 

32 
33 

34 

.24561 
.24601 
.  24642 
.  24682 

46 
46 

•  32970 
.33019 
.33069 
.33118 

y 
49 
49 
49 

AC\ 

.  26964 
.27003 
.27042 
.27082 

39 
39 
39 
39 

•?r> 

•35904 
•35952 
.36001 
.36049 

4o 

48 
48 
48 

AO 

32 
33 
34 

6 
7 
8 

46 

4  o 
4-7 
5-4 

4°o 

4-$ 
5-3 

P 

37 
38 

9.24723 
.24764 
.  24804 
.24845 

41 
46 
46 
46 

9.33I67I 

.332I61 
.33266 

•33315  j 

49 
49 
49 
49 

9.27I2I 
.27161 
.27200 
.27239 

39 
39 
39 
39 

9.36098 
.36145 
.36194 
•36243 

48 
48 
48 

48 

AQ 

35 
36 
37 
38 

9 
10 

20 
30" 
40 
50 

6 

13 
20 

27 

•  7 

.0 

•7 

6.0 
6.* 

*3-3 
20.  o 
26.  £ 
33-3 

39 

.24885 

4^ 

•  33364 

49 

.27273 

39 

.36291 

48 

AO 

39 

40 

42 
43 
44 

9.24926 
.24966 
.25007 

.25047 
.25087 

40 
40 
46 
46 
40 

An 

9.334I3 
.33463 
.33512 
.33561 
.33610 

49 
49 
49 

3 

9.27318 
.27357 

.27396 
.27435 
•27475 

39 
39 
39 
39 
39 

9-36340 
.36388 

.  36436 
.  36484 

.36533 

48 
48 

48 
48 

48 

AO 

40 

42 
43 
44 

6 

7 
8 
9 

39 

3-9 
4.6 
5-2 

39 

3-9 
4-5 
5-2 

41 
46 

47 
48 

9.25128 
.25163 
.25209 
.25249 

4O 
40 
40 

40 

9-33659 
•33708 
.33758 
.33807 

49 
49 
49 
49 

9.275H 

•27553 
•27592 
.27631 

39 

3? 
39 
39 

9.36581 
.36629 
.36678 
.36726 

48 
48 
48 
48 

11 

47 
48 

10 

20 
30 
40 

6 
13 
19 
26 
32 

6 
i 
7 
3 
9 

13-0 
19.5 
26.0 
32-5 

49 

.25289 

4 

.33856 

49 

.27676 

39 

.36774 

10 

49 

50 

52 
53 

9.25329 
.25370 
.25410 
.25456 

40 
40 
40 
46 

9.33905 

•33954 
•34003 
.34052 

49 
49 
49 
49 

9.27709 

•27749 
.27788 
.27827 

3? 
39 
39 
39 

9.36822 
.36876 
.36919 
-36967 

48 
48 
48 
48 

50 

52 
53 

6 
7 

38 

3-8 
4-5 

54 

.25490 

ir» 

.34101 

49 

.27866 

39 

.37015 

AQ 

54 

9 

5-1 
5-8 

55 

9.25531 

4O 

9-34I50 

49 

9.27905  ^y 

9.37063 

4« 

55 

10 

6.4 

56 
57 
58 

.25571 
.25611 
.25651 

4U 

40 

40 

•34199 
.  34248 

•  34297 

49 
49 
49 

•27944  ^ 
.27982!  38 
.28021;  39 

.37111 

.37159 
.3720? 

48 
48 

48 

56 

3° 
40 
5° 

19.2 
25-6 
32-1 

59 

.25691 

4° 

AO 

•34346 

49 

J.Q 

-28066  39 

.37255 

48 

59 

60 

9.25731 

9-34395 

9.28099  ** 

9.37303 

60 

LOST.  Vers.   /> 

Log.  Ex  «ec. 

/>   Los.  Vers.   J> 

tag.  Exser.   J> 

P. 

P. 

411 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

36°  37° 


/ 

Log.  Vers. 

Log.  Kxsec. 

Log.  Vers. 

jog.  Exsec.j  -D 

/ 

p.  P. 

0 

I 

2 

9.28099 
.28138 
.28177 

39 
38 

9.37303 
•37352 
.37400 

48 
48 

,,0 

9-30398 
.30436 
•  30474 

3? 
3? 

9.40163 
.4O2IO 
.40258 

47 

4? 

0 

i 

2 

3 

.28216 

39 

.37448 

40 

/|Q. 

.30511 

37 

•40305 

47 

Afj 

3 

48    48 

4 

.28255 

39 

no 

.37496 

40 

.0 

•  30549 

np. 

•40352 

47 

4 

6 

4'8 

4.8 
c  5 

5 

9.28293 

38 

9-37544 

45 

9.30587 

5 

9.40399 

47 

5 

8 

M 

1 
6.4 

6 

7 

•28332 
.28371 

39 
38 

•37592 
•  37640 

48 

A*} 

.30624 
.30662 

37 

.40447 
.40494 

47 

6 

7 

9 

10 

20 

ie!i 

7-2 

8.0 
16.0 

8 
9 

.28410 
.  28448 

39 
38 

.37687 
•37735 

47 
48 

A  9. 

.  30700 
.3073? 

37 

.40541 
.40583 

47 
47 

8 
9 

3° 
4° 
50 

24.2 

32.3 
40.4 

24.0 
32.0 
40.0 

10 

9.28487 

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9.37783 

45 

,0 

9.30775 

nfl 

9.40635 

47 

10 

1  1 

12 
13 
H 

.28526 
.28564 
.28603 
.  28642 

38 
38 
39 
38 

•3783? 
.37879 
.3792? 
•37975 

40 
48 
48 

4? 

.  0 

.30812 
.30850 
.3088? 
•30925 

37 
3? 
3? 
3? 

n<? 

.40682 
•4073° 
•40777 
.40824 

47 

47 
47 

1  1 

12 

13 
U 

6 
7 

4-Z 

5-5 

47 

4-7 

5-5 

15 

16 

9.28686 
.28719 

no 

38 

9.38023 
.38071 

48 

48 

9.30962 
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37 
3? 

9.40871 
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47 
4? 

15 
16 

8 
9 
10 

6-3 
7-1 
7>9 

6.2 

7.6 
7-8 

17 
18 

19 

.2875? 
.28796 

.28835 

39 

.38119 
.38165 
.38214 

4? 
48 

A  5 

.3103? 
.31075 
.31112 

3? 

n<7 

.40965 
.41012 
.41059 

47 
47 
47 

17 
18 

19 

20 

3° 
40 
50 

I:! 

39-6 

'5-6 
23-5 

20 

21 
22 
23 

.28912 
.28950 
.28983 

38 
38 
38 
38 

•20 

9.38262 
.38310 
.3835? 
.38405 

47 
48 
4? 
48 

9.3II50 
.31187 
.31224 
.31262 

37 
37 
3? 
3? 

9.41105 

•4II53 
.41206 

.4124? 

47 
47 
47 
47 

20 

21 
22 
23 

46 

6     4-6 

24 

.20027 

38 

.38453 

.  0 

.31299 

37 

_c 

.41294 

47 

24 

7     5-4 
8     62 

27 
28 

9.29065 
.29104 
.29142 
.29186 

no 

38 
38 

II 

•20 

9.38501 
•38548 
.38596 
.38644 

48 
4? 
48 

4? 

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9-31336 
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.31411 

•  3H48 

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i 

9.4I34I 
.41388 
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47 
47 
47 
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25 
26 

27 
28 

9     7.0 
10     7.7 

20      I5.S 

30   23.2 
40   31.0 

29 

.29219 

38 

•  38692 

45 

fl 

.3H85 

37 

n<7 

.41529 

47 

29 

5°   38.7 

30 

32 

9.29257 
.29295 
.29334 

no 

38 
38 
38 

9-38739 
.38787 
.38834 

47 
4? 
4? 

9.31523 
.31560 
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3? 
37 
37 

9.41576 
.41623 
.41670 

46 
47 
47 

30 

32 

39    38 

33 
34 

.29372 
.29410 

38 

.38882 
•  38930 

4? 

A  <T 

•31634 
.31671 

3? 
37 

.41717 
.41763 

47 
46 

33 
34 

6 

I 

3-9 
4-5 

5-2 

3-8 
4.5 

11 

9-29448 
.29487 

38 
38 

nO 

9.38977 
.39025 

47 
4? 

9-3I708 
.31746 

5 

9.41816 
.4185? 

47 
47 

3 

9 

10 

20 

1:1 

13.0 

5-8 
6.4 

12.  § 

39 

.29525 
.29563 
.  29601 

3° 

38 

38 

-0 

.39072 
.39126 
.39168 

48 
4? 

o 

•31783 
.31820 

-31857 

37 
37 
37 

.41904 
.41951 
.41998 

46 

47 
47 

37 
38 
39 

3° 
40 

19.5 
26.0 
32.5 

19.2 
25-6 

3'.  i 

40 

42 

9.29639 
.2967? 
.29715 

38 

38 

3? 

9.39215 
.39263 
.39310 

47 
48 

4? 
Aa 

9.31894 
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37 
37 
37 

9.42044 
.  4209! 
.42138 

46 
47 
46 

40 

42 

g 

38 

31 

43 

.29754 

nQ 

.39358 

47 

•32005 

37 

.42185 

47 

43 

4-4 

4.4 

44 

.29792 

3° 

ntt 

.39405 

Ao 

.  32042 

37 

.42231 

46 

44 

8 

5-° 

c  g 

45 
46 
47 
48 

49 

9.29830 
.29868 
.29906 

.29944 
.29982 

3° 
38 
38 
38 
38 

9-39453 
•39506 
.39548 

•39595 
.39642 

47 
4? 
4? 
47 
4? 

A  $ 

9-32079 
.32116 

•32153 
.32190 

.32227 

37 
37 
37 
37 
37 

nP 

9-42278 
•42325 
.42372 

.42418 
.42465 

47 
46 
47 
46 
47 

45 
46 

47 
48 
49 

10 

20 

3° 
40 
So 

I2.g 

19.0 
25-3 
31-6 

6.2 

12.5 

18.7 

25.0 
31-2 

50 

9  .  30020 

3^ 

9-39690 

47 

9.32263 

36 

9.42512 

46 

50 

52 
53 

54 

•3005? 
.30095 

.30133 
.30171 

38 
38 
38 

•3973? 
•39785 
•39832 
.39879 

4? 
47 
4? 

A  <9 

.32300 
•3233? 
.32374 
.32411 

37 
37 
36 
37 

•42558 
.42605 
.42652 
.42698 

46 

47 
46 
46 

52 

53 
54 

6 

I 
9 

37 

3-7 

4'3 
4.9 

5-5 

36 

3.6 
4.2 

4-8 
5-5 

11 

9.30209 
.30247 

3? 

9-39927 
•39974 

47 

47 

Afj 

9.3244? 
•  32484 

36 

? 

9.42745 
•42792 

46 
47 

55 
56 

10 

20 
30 

6.1 

12.3 
18.5 

6.1 

12.  I 

lS.2 

11 

.30285 
.30322 

3? 

.40021 
.  40069 

47 
4? 

.32521 
•32558 

36 

37 

.42838 
.42885 

46 
46 

A2 

57 

58 

40 
5° 

24-6 
30-8 

24.3 
30.4 

59 

.30366 

38 

.40116 

47 
4? 

.32594 

36 

•27 

.42931 

46 

Af. 

59 

(JO 

9-30398 

9.40163 

9-32631 

9.42978 

60 

Log.  Vers 

D 

Log.  Kxsec.  J> 

Log.  Vers. 

7> 

Loir.  Kxsec.   /> 

' 

P.  P. 

412 


TABLE   VIII.— LOGARITHMIC   VERSED   SINES    AND    EXTERNAL   SECANTS. 

38°  39° 


1'  / 

Log.  Vers. 

D 

Log.  Exsec. 

D  I 
" 

Log.  Vers. 

1) 

Log.  K\MT. 

i> 

P.  P. 

0 

9.3263? 

. 

9.42978 

9.34802 

.3? 

9-45752 

g 

0 

I 

2 

3 

4 

.32668 
.32704 

.32741 
.32778 

36 

3 

-2 

.43024 

.43071 
.43118 
.43164 

47 

.  2 

.3483? 
,  .34873 
.34909 

•34944 

i 

35 

_  a 

•4579? 
.45843 
.45889 

-45935 

46 
46 

46 

,  p 

i 

2 

3 
4 

6 

47 

4-7 

46 

4-6 

I 

8 
9 

9.32814 
.32851 
.32888 

.32924 
.32961 

§ 

37 

*>2 

9.432II 
•4325? 
.43304 
.43350 

•43396 

46 
46 
46 

3 

A2 

9.34980 
.35016 
.35051 
•35087 
.35122 

35 
36 

11 

35 
_  P 

9.45981 
.46027 
•46073 
.46118 
.46164 

45 
46 

46 
45 
46 
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6 

8 
9 

8 
9 

10 

20 
30 
40 
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7-1 
15-6 
23.5 

6.2 

4? 

23-2 

3^-0 
38.7 

10 

ii 

9.32997 
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i 

9-43443 
.43489 

46 
46 
46 

9-35158 
.35193 

35 

35 
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9.46216 
.46256 

40 

4I 

Ah 

10 

ii 

12 
13 

14 

.33070 
.33107 
.33143 

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.43536 
.43582 
.43629 

46 
46 

.35229 
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•35300 

35 

_  p 

.46302 
.4634? 
.46393 

45 

46 

.  p 

12 
13 
H 

6 
7 

46 

4.6 
5-2 

4S 

4-5 
5-3 

II 

9.33180 
.33216 

36 

1 

9-43675 
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46 
46 

9-35335 
•35376 

35 

35 

9-46439 
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45 

46 

!i 

8 
9 

10 

6.1 
6.9 

7-6 

6.6 
6.8 

7.6 

17 

18 
19 

•33252 
.33289 
.33325 

36 
36 
36 

.43768 

.43814 
.43861 

46 
46 
46 
\f\ 

.35406 
•35441 

•35477 

35 

11 

-46536 

•46576 
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45 

17 

18 
19 

20 

3° 
40 

50 

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23.0 

22.7 
30-3 
37-9 

i  20 

9-3336I 

36 

o2 

9-43907 

40 

9-35512 

5 

9.46668 

20 

21 

22 

.33398 
•33434 

36 
36 

l6 

•43953 
.43999 

46 
46 

A2 

-3554? 
.35583 

35 
35 

•46713 
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4s 

21 

22 

45 

23 

•33470 

3O 

.44046 

46 

.35618 

5 

3? 

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Ap 

23 

6 

24 

-33507 

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•44092 

.  2 

-35653 

i 

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.46856 

45 

A  ? 

24 

7 
8 

I:! 

25 

9-33543 

3° 

9-44138 

46 

A2 

9.35689 

35 

9.46896 

45 

25 

9 

6.7 

26 

27 

•33579 

JO 

£ 

.44185 
•44231 

46 
46 

-35724 
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35 
35 

.46942 
•4698? 

45 

26 

27 

20      I 
30      2 

7-5 
50 
2.5 

28 
29 

.33652 

.33688 

$ 

.44277 
.44323 

46 
46 

•35794 
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35 
35 

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•47078 

45 
45 
A£\ 

28 

29 

40    3 
50    3 

o.o 
7-5 

80 

9.33724 

36 

9-44370 

46 

9.35865 

35 

9.47124 

40 

30 

31 

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A2 

.35900 

35 

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.47170 

Ap 

31 

32 

•33796 

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.44462 

46 

Aft 

•35935 

5 

.47215 

45 

Ap 

32 

37 

36 

33 

34 

•33833 
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36 

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.44508 

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4° 
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-35970 
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35 
35 

.47261 
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45 
45 

Af\ 

33 
34 

6 

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3-7 
4-3 
4.9 

11 

4-8 

1 

39 

9.33905 
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36 

36 
36 
36 

9.44601 

.44647 
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•44739 
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46 
46 
46 

46 
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9  .  36046 
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.36146 
.36181 

35 
35 
35 
35 
35 

9-47352 
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•47443 
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40 

45 
45 
45 

45 

A  8 

35 
36 

39 

9 
10 
20 
30 
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1:1 

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24-6 
30.3 

12.  I 

18.2 

24.3 
30.4 

40 

9.34085 

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9.44831 

4O 
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9.36216 

35 

9.47580 

45 

40 

41 

42 

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36 

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46 

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35 

35 

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45 

42 

6 

36 

3.6 

35 

3.5 

43 
44 

•34193 
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36 

.44970 
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4U 
46 

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.36321 
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35 

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.47762 

45 

43 

44 

I 

g 

4.2 

4.8 
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5.3 

45 

46 

9.34265 
•34301 

36 

9.45062 
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Afi 

9-36391 
.  36426 

35 
35 

9.47807 
.47852 

45 
45 

AS 

45 
46 

10 
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3° 

6.0 

12.  O 

18.0 

5-9 

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17.7 

47 
48 
49 

•34337 
•34373 
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1 

.45154 
.45200 
.45246 

40 
46 
46 

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•  3646  1 
.36495 
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35 
34 

35 

•47898 
•47943 
•47989 

43 

45 
45 

A  F. 

47 
48 

49 

40 
50 

24.0 
30.0 

23  6 
29.6 

50 

51 
52 

9-34444 
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36 
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9.45292 
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46 
46 

46 
Af. 

9.36565 
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35 
35 
34 

9.48034 
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45 
45 

50 

52 

6 

35 

35 

34 

3- 

53 

54 

-34552 
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jv 

35 

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4O 

46 
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.36670 
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35 
35 

.48176 
.48216 

45 

A  r 

53 

54 

7 
8 
9 

tj 

5>2 

4- 
4* 

5- 

55 
56 

57 
58 
59 

9-34623 
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.34695 
•34736 
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35 

36 

II 

9.45522 
.45568 
.45614 
.  45660 
.45706 

46 
46 
46 
46 
46 

4.6 

9-36739 
.36774 
.36809 

.36844 
.36878 

34 

II 

35 

34 

•3C 

9.48261 
-48306 
•48352 

.4839? 
.48442 

45 
45 
45 
45 
45 

A? 

i 

59 

20 
30 
40 

5° 

.f:S 

'7-5 
23-3 
29-! 

5- 
ii. 

23- 

28. 

60 

9.34802 

9-45752 

9-36913 

9.48488 

60 

* 

Log.  Vers. 

7> 

Lotf.  Exsec. 

7> 

Loar.  Vers. 

I) 

' 

1'.  I'. 

413 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL    SECANTS. 

4O°  41° 


/ 

Los?.  Vers. 

J> 

jog.  Exsec.   D 

Log.  Vers. 

LOST.  Exsec. 

D 

• 

p.  P. 

0 

9  .369!  3 

9 

9.48488 

9.38968 

9.51190 

0 

I 

2 

3 

.36948 
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34 

34 
35 

.48533 
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45 
45 
45 

.39002 

.39035 
.39069 

34 
33 
34 

•51235 
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45 
44 
45 

I 

2 

3 

4 

.37052 

34 

.48669 

45 

A  £ 

.39103 

33 

•51369 

44 

4 

45    45 

5 

9-37086 

34 

9.48714 

45 

9.39137 

34 

9.51414 

45 

S 

6 

4-5 

4-5 

6 

.37121 

35 

.48759 

45 

.39170 

33 

35 

•5-1458 

44 

6 

7 
8 

6.6 

S-2 
6.0 

8 
9 

.37156 
.37196 
.37225 

34 
34 
34 

.48805 
.48850 
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45 
45 
45 

.  39204 
.39238 
.39271 

3 
34 
33 

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•51503 
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45 
44 
44 

7 
8 

9 

9 
10 

20 
30 

6.8 
7.6 
15.1 
22.7 

6.7 
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15-0 
22.5 

10 

ii 

12 

9-37259 
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34 
34 
34 
34 

9.48946 
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45 

45 
45 

45 
4? 

9.39305 

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33 
34 
33 

33 

1% 

9.5163? 
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45 
44 
44 
45 

A'  A 

10 

1  1 

12 
13 

40 

5° 

3°-3 
37-9 

37-5 

H 

.3739? 

34 

.49121 

5 

•  39439 

33 

.51816 

44 

15 

16 

9-37432 
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34 

II 

9.49166 
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45 
45 
4? 

9-39473 
•  39507 

33 

9.51  866 
.51905 

44 
45 

AA 

II 

6 

44 

4-4 

44 

4-4 

17 
18 

•37501 
•37535 

34 

1A. 

.49257 
.49302 

45 

A  r 

.39540 

•39574 

33 

•35 

.51950 
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44 

AA. 

17 

18 

7 
8 
9 

5-2 

5-9 
6.7 

5-8 
6.6 

T9 

•37570 

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.49347 

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•  3960? 

JJ 

.  52039 

19 

10 

7-4 

20 

21 

22 

9.37604 
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34 

34 

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9.49392 
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.49482 

5 
45 
45 

9.39641 

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33 
33 

9.52084 
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45 
44 
44 

20 

21 

22 

30 
40 
50 

14-8 

22.2 
29.g 

37-1  . 

22.0 

36*6 

23 

.3770? 

34 

•4952? 

45 

•39741 

33 

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44 

1  A 

23 

24 

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34 

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45 

A  ? 

.39774 

33 

<•>  5 

.52262 

44 

24 

y 

27 
28 

29 

9.37776 
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34 
34 
34 
34 
34 

9.49618 
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•49753 
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45 
45 
45 
45 
45 

9.39808 

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33 
33 
33 
33 
33 

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9-52305 
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.  52440 
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44 

45 
44 
44 
44 

27 
28 
29 

6 

7 

8 

35 

3-5 
4.1 

4  6 

34 

3-4 
4.0 
4.6 

80 

9-3794? 
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34 
34 

9-49843 
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45 
45 

9-39975 
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33 

33 

9-52529 

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44 
44 

80 

9 

10 

20 

5-2 
ii.  g 

5-7 

32 
33 

.38016 
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34 
34 

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45 
45 

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33 

35 

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44 
44 

32 
33 

3° 
40 

50 

17-5 
23-3 
29.1 

17.2 
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28.7 

34 

•  38084 

34 

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45 

.40108 

3 

.  5270? 

44 

34 

35 

9-38II8 

34 
i/f 

9.50068 

45 

9.40141 

33 

35 

9.52752 

44 

35 

36 

11 

39 

•38153 
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34 
34 
34 
34 

.50113 
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45 
45 
45 
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3 
33 

33 
33 

.  52796 
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44 
44 
44 
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36 

37 
38 
39 

6 

34 

3-4 

33 

3-3 

40 

42 
43 

9.38289 
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34 
34 
34 
34 

9-50293 
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45 
45 
45 
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9-4030? 
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33 
33 

11 

9-52974 
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.  53063 
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44 
44 
44 
44 

40 

42 

43 

7 
8 
9 

10 

20 

3-9 
4-5 
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5  6 
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17.0 

3-? 
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16.7 

44 

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34 

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45 

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33 

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44 

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22.6 

28.5 

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27.  Q 

4I 
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9-38459 
•38493 

34 

34 

9-5051? 
.  50562 

45 
45 

9.40473 
.40505 

33 
33 

•3? 

9.53196 
•  53246 

44 

44 

45 
46 

47 

.3852? 

34 

.5060? 

45 

.40540 

33 

•53285 

44 

47 

48 
49 

.38561 
.38595 

34 
34 

.  50652 
.  50697 

44 

45 

•40573 
.  40606 

33 
33 

.53329 
.53374 

44 
44 

48 
49 

50 

9.38629 
.38663 

34 
34 

9.50742 
.50787 

45 

45 

9.40639 
.40672 

33 
33 

9.53418 
.53462 

44 
44 

50 

Si 

33 

6     3-3 
7     3-8 

52 

.38697 

34 

.  50831 

44 

.40705 

33 

.53507 

44 

S2 

8     4.4 

53 

.38731 

33 

•50875 

45 

.40738 

33 

•53551 

44 

S3 

10     5.5 

54 

.38765 

34 

.  50921 

45 

.40771 

33 

•53595 

44 

54 

2O       1I.O 

55 
56 
57 
58 

9.38799 
•38833 

'38906 

34 
34 
33 
34 

9.50966 
.51011 

•51055 
.51106 

44 
45 
44 
45 

9  .  40804 
.40837 
.  40870 
.40903 

33 
33 
33 
33 

9.53640 
•53684 
•53728 
•53773 

44 
44 
44 

44 

55 
56 
57 
58 

40       22.  O 
5°       27.5 

59 

.38934 

33 
•34 

.51145 

45 

AA 

.40936 

•3-3 

.53817 

44 

AA 

59 

60 

9.38968 

9.51190 

9.409^9 

9.5386! 

<>0 

' 

Lop.  Vers.   7> 

joar.  Kxser.   /> 

Loir.  Vers.   /> 

L.«r.  Kxs-r.   7> 

...' 

P.  P. 

414 


TABLE    VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

43°  43° 


/ 

Log.  Vers. 

J> 

josr-  Exsec. 

Log.  Vers. 

.MI:.  Exsec. 

J> 

/ 

P. 

1>. 

0 

I 

2 

3 
4 

9.40969 
.41001 
.41034 
.4106? 
.41106 

32 

33 
33 
33 

-a 

9.53861 
.53906 
•53950 

•  53994 
•  54038 

4? 
44 
44 
44 

9.42918 
.42950 
.42982 
.43014 

•43046 

32 
32 
32 
32 

9.56505 
•  56549 
•56|93 
.56637 
.  56686 

43 
44 
44 
43 

0 

i 

2 

3 
4 

At 

f 

6 

8 
9 

9-4II33 
.41166 
.41199 
.41231 
.41264 

32 

33 
33 
32 
33 

9.54083 
.54127 
.54171 
.54215 
.54259 

44 
44 
44 
44 

9-43°78 
.43116 

.43U2 
.43174 
.43206 

32 

£ 

32 

32 

9.56724 
.  56768 
.56812 
.56856 
.56899 

44 
44 

43 
44 
43 

6 

8 
9 

6 
7 
8 
9 

10 
20 

4* 

4- 

5- 

1: 

7- 

14. 

4 
1 

3 

7 

44 

4.4 

1:1 

7-3 
M-6 

22  O 

10 

ii 

12 
13 

H 

9.41297 
.41330 
.41362 
.41395 
.41428 

33 
32 
33 
32 

9.54304 
•  54348 
•54392 
.  54436 
.  54486 

44 
44 
44 
44 
44 
.  'f 

9.43238 
.43270 
•43302 
•43334 
.43365 

32 
32 
32 

s 

9  '.5698? 
.57031 
•57075 
•57"8 

44 
44 
43 
44 

43 

10 

ii 

12 
13 
14 

40 

5° 

37- 

15 

16 

17 
18 

19 

9.41461 

•4*493 
•41526 
.41559 
.41591 

33 
32 

!i 

32 

9.54525 
.54569 
•54613 
•  54657 
•  547oT 

44 
44 
44 
44 
44 

9-4339? 
.43429 
.43461 

-43493 
.43525 

32 
32 
32 

32 

9.57162 
.57206 
.57250 
.57293 

•5733? 

44 
43 
44 
43 
44 

15 
16 

17 
18 

19 

6 

I 

9 

4: 

4- 
6. 

\ 
j 

43 

4-3 

5-7 
6.4 

20 

21 

22 
23 
24 

9.41624 
.41657 
.41689 
.41722 
.41754 

32 

32 
32 

9-54745 
-54790 
.54834 
.54878 
.54922 

44 

44 

9-43557 
-43588 
.43626 
.43652 
.43684 

3? 
32 
-y 

•  57424 
•  57468 
.57512 
•57556 

43 
43 
44 
43 
44 

20 

21 

22 

23 
24 

20 

3° 
40 

50 

14. 

21. 
29- 
36. 

j 

2 

21.5 

28.6 
35-8 

25 
26 

27 
28 

29 

9.41787 
.41819 
.41852 
.41885 
.41917 

3? 

33 
32 

9.54966 
.55016 

•55054 
•55098 
.55U2 

44 

9.437I5 
•43747 
•43779 
.43810 

.43842 

31 
32 
3i 
3i 
32 

9-57599 
•  57643 
•  57687 
•  57730 

•57774 

43 
43 
44 
43 
43 

25 
26 

27 
28 
29 

6 

7 

3, 

3- 
3- 

3 

a 

?| 

30 

32 
33 
34 

9.41950 
.41982 
.42014 
.42047 
.42079 

2 
32 
32 

9.55186 
•5523o 
•55275 
•55319 
.55363 

44 

9.43874 
.43906 

•4393? 
•43969 
.44006 

1 

3i 

3? 

9.578.8 
•5786! 
•57905 
•  57949 
•  57992 

44 
43 
43 
44 
43 

30 

32 
33 
34 

9 
10 

20 

3° 
40 
50 

4> 
5- 
ii. 
16. 

22. 

27. 

9 
5 

0 

5 

0 

5 

4-3 
4-9 
5-4 
10.  § 
16.2 

27.1 

35 
36 

39 

9.42112 
.42144 

.42177 
.42209 
.42241 

1 

32 
32 

9-55407 
•55451 
•55495 

44 
44 

tt 

9.44032 
.44064 

.44095 
.44127 
•44158 

32 
3? 

I 

9.58036 

•58079 
.58123 
.58167 
.58216 

43 
43 
44 

43 
43 

.s 

35 
36 
37 
38 
39 

3 

2 

31 

40 

42 
43 
44 

9-42274 
•42306 
•42338 
.42371 
.42403 

32 

32 
3? 

32 

9.55627 
.55671 
.55715 
.55759 
•55803 

44 
44 
44 
44 
44 

9.44190 
.44221 

.44253 
.44284 
.44316 

3* 

3i 

9.58254 
.5829? 
•5834? 
•58385 
•58428 

43 
43 
44 
43 
43 

40 

42 
43 
44 

7 
8 
9 
10 

20 
30 
40 

3- 
4- 
4* 
5- 
10. 
16. 

21  . 

j 
8 

0 

3-7 

4.  9. 
5-2 

10.5 

15.7 

21  .O 

45 
46 
47 
48 
49 

9-42435 
•4246? 
.42500 
-42532 
.42564 

3: 
32 

9.55847 
.55890 
•55934 
•55978 
.56022 

44 
43 
44 
44 
44 

9-44347 
.44379 
.44416 

•44442 
•44473 

31 

$ 

3? 

9Jsti5 

•58559 
.58602 
.  58646 

43 
43 
43 
43 
43 

45 
46 
47 
48 
49 

5° 

26.2 

50 

51 

52 
53 

54 

9-42596 
.42629 
.42661 
.42693 
.42725 

32 
32 
32 
32 
32 

9.56065 
.56116 

.56154 
.56198 
.  56242 

44 
44 
44 
43 
44 

9.44504 
•44536 
.4456? 
.44599 
-44630 

31 

S 

3i 
3i 
tf 

9.58689 
•58733 
•58776 
.  58826 
.  58864 

43 
43 
43 
44 
43 

50 

52 
53 
54 

6 

7 
8 

20 

I 

31 

3-i 

5:1 

J:l 

°.§ 

55 

56 
57 
58 
59 

•'42789 
.42822 
.42854 
.42886 

32 
32 
32 

32 
32 

7,2 

9.56286 
•'56330 
.56374 
•5641? 
.  5646! 

44 
44 
44 
43 
44 

4.5 

9.4466! 

.44693 
.44724 

•44755 
•44787 

31 
3i 
3i 
3i 
3i 

9.58907 
.58951 
•  58994 
•5903? 
.59081 

43 
43 
43 

43 

55 
56 
57 
-58 
59 

3° 

40 

50 

2 
2 

°-§ 
5-8 

60 

9.42918 

9-56505 

9  .  448  i  8 

9.59124 

J 

00 

* 

Log.  Vers. 

J> 

LOST.  Exsec. 

J> 

Loir.  Vers. 

7> 

jOjr.  Ex  sec'. 

j) 

p. 

1'. 

415 


TABLE   VIIL— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

44°  45° 


' 

Log.  Vers 

j> 

Log.  Exsec. 

I) 

Log.  Vers. 

_D 

Log.  Exsec. 

D 

' 

p.  P. 

0 

2 

3 

4 

9.44818 
.44849 
.44886 
.44912 
•44943 

3? 

| 

i? 

9.59124 
.59168 
.59211 
.59255 
•  59298 

43 
43 
43 
43 

9.46671 
.  46701 
.46732 
.46762 
•46793 

30 
30 
30 
30 

9.61722 
.61765 
.61808 
.61852 
.61895 

43 
43 
43 
43 

0 

I 

2 

3 

4 

43 

I 

7 
8 

9 

9.44974 

•45036 
.45068 

.45099 

31 

9-59342 

•59385 
.  59429 

•59472 
.59515 

43 
43 
43 
43 

43 

.  % 

9.46823 
.46853 
.46884 
.46914 
.46945 

30 
30 
30 
30 

30 
,~ 

9.61938 
.61981 
.  62024 
.6206? 
.62110 

43 
43 
43 

43 
43 

6 

8 
9 

6     4-3 
7     5-i 
8     5.8 
9     6-5 
10     7.2 
20    14.5 
30    21.7 

4-3 
14-3 

10 

n 

12 
13 

14 

9.45130 

.45192 

.45223 
.45254 

31 

S| 

9-59559 
.59602 

.59646 
.59689 
.59732 

43 
43 
43 
43 
43 

9.46975 
.47005 
.47036 
.47065 
.47096 

3° 
30 
30 
36 
30 

9.62153 
.62195 
.62239 
.62282 
.62326 

43 
43 
43 
43 
43 

10 

ii 

12 

13 
U 

40    29  .  o 
50    36.2 

28.fi 
35-8 

15 

16 

17 
18 

19 

9.45285 
•45316 
.45348 

•45379 
.45410 

31 

9.59776 

.59819 
.59863 
.  59906 

.  59949 

43 
43 
43 
43 

43 
a 

9.47127 

.4715? 
•47l8? 

.47218 
.47248 

3° 
36 
30 
36 
30 

9.62369 
.62412 
.62455 
.62498 
.62541 

43 
43 
43 
43 
43 

15 

16 

17 
18 

19 

6 
9 

10 

42 

4.2 
4.9 
5-6 
6.4 

20 

21 

22 

23 

24 

9.45441 
.45472 
.45503 
•45534 
.45565 

31 

_~ 

9-59993 
.  60036 
.60079 
.60123 
.60166 

43 
43 
43 
43 
43 

9-47278 
•47308 
•47339 
.47369 
•47399 

36 
30 
36 
30 
36 

9.62584 
.62627 
.62670 
.62713 
.62756 

43 
43 
43 
43 
43 

20 

21 

22 
23 
24 

30      2 
40      2 
50      3 

4.1 

1.2 

8-3 

5-4 

25 
26 

27 
28 

29 

9.45595 

•4565? 
.4568§ 
.45719 

3° 

9.60209 

.  60296 

•60339 
.60383 

43 
43 
43 
43 
43 

9.47429 

•47459 
•47490 
.47520 
•47550 

3° 
30 
36 
30 
30 

9.62799 
.62842 
.62885 
.62928 
.62971 

43 
43 
43 
43 
43 

25 
26 

27 
28 

29 

7     3-7 
8     4.2 

31 

3-1 
3-6 
4.1 

30 

32 
33 
34 

9.45750 
.45781 
.45812 
.45843 
.45873 

31 
30 

30 

9.60426 
.  60469 
.60512 
.60556 
.60599 

43 
43 
43 
43 
43 

9.47586 
.47616 
.47646 
.47676 
•47700 

30 
30 
30 
30 

9.63014 
.63057 
.63100 

.63143 
.63186 

43 
43 
43 
43 
43 

30 

32 
33 
34 

9     4.7 
10     5.2 

20      T0.5 

3°      15-7 
40      21  .O 

50      26.2 

4*1 

10.3 

20.6 

25-8 

35 
36 

P 

39 

9.45904 

•4593| 
.45966 

•45997 

.4602? 

31 
33 
30 

9.60642 
.60685 
.60729 
.60772 
.60815 

43 
43 
43 
43 
43 

9-  4773  i 
.4776i 

-47791 
.47821 

.47851 

30 
3° 
30 
30 

9.63229 
.63272 
.63315 
.63358 
.63401 

43 
43 
43 
43 
43 

11 

37 
38 
39 

36 

6     3.0 

30 

40 

41 
42 

43 

44 

9.46053 
.46089 
.46120 
.46156 
.46181 

30 

9.60858 
.  60902 
.60945 
.  60983 
.61031 

43 
43 
43 
43 
43 

9.47881 
.47911 
.47941 

•47971 
.48001 

30 
30 
30 
30 
30 

9  :  63485 
.63529 
.63572 
.63615 

42 
43 
43 
43 
43 

40 

42 
43 

44 

I  1:1 

9     4.6 
10     5.1 

2O       IO.I 

30    15-5 
40    20.3 
50    25.4 

3-5 
4.0 
4-5 
5-o 

IO.O 

15-0 

20.  o 
25.0 

45 
46 

47 
48 

49 

9.46212 
.  46242 
.46273 
.46304 
.46334 

30 

33 
30 

9.61075 
.61118 
.61161 
.61204 
.6124? 

43 
43 
43 
43 

43 
.  ~ 

9.48031 
.48061 
.48096 
.48126 
.48156 

3° 
30 
29 
30 
30 

.63701 

.63744 
.63787 
.63830 

43 
45 
43 
43 
43 

45 
46 

47 
48 

49 

_" 

50 

52 
53 

54 

9.46365 
.46396 
.46425 
.46457 
.4648? 

3r 
30 

36 
30 

9.61291 
.61334 
•6i377 
.61426 
.61463 

43 
43 
43 
43 
43 

9.48186 
.48216 
.48240 
.48270 
.48300 

3° 
30 
29 
30 
30 

•63915 

•63958 
.64001 
.64044 

43 
42 

43 
43 
43 

.a 

50 

52 

53 
54 

6 
7 

8 
9 

10 

20 

29 

3-9 
4-4 
*-9 
?•§ 

55 
56 

57 
58 
59 

9.46513 
.46549 
.46579 
.46610 
.46646 

31 
30 
36 
30 
3° 

9.61505 
.61550 

•61593 
.61636 
.61679 

43 
43 
43 
43 
43 

9.48329 

•48359 
.48389 
.48419 
.48449 

29 
30 
30 
29 
30 

9  .  64087 
.64130 
.64173 
.64216 
.64258 

42 

43 
43 
43 

42 

AT. 

59 

40    i 

50      2 

>*| 

3-6 

l-o 

60 

9.46671 

9.61722 

9  -48478 

9.64301 

60 

' 

Log.  Vers. 

j> 

Log.  Kxsec. 

y> 

Log.  Vers. 

5 

Log.  Exsec. 

I) 

' 

F.  P. 

416 


TABLE    VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

46°  47° 


/ 

Log.  Vers. 

D 

josr.  Exsec.  1> 

Loe.  Vers. 

J> 

Logr.  Exsec.  |  1> 

/ 

P.  P. 

0 

9.48478 

9-6430I   ,- 

9.50243 

9.66864 

^ 

0 

2 

.48508 
.48538 

3° 
29 

.64344 
.64387 

4J 
42 

.50272 
.  50301 

29 
29 

.66907 
.66950 

42 

43 

/12 

i 

2 

3 

4 

.48568 
.4859? 

3° 
29 

•6443° 
.64473 

43 
43 

1a 

•  50330 
.50359 

29 
29 

.  66992 
.67035 

42 
42 

3 

4 

6 

9.4862? 
•48657 

3° 
29 

9.64515 

-64558 

42 

43 

9.50388 
.50417 

29 
29 

9-6707? 
.67120 

42 

4? 

6 

43 

6     43 

42 

4.2 

8 
9 

.48685 
.48716 
.48746 

30 
29 

_a 

.64601 
.64644 
.64687 

3 
42 
43 

.50446 
.50475 

29 
29 
29 

.67162 
.67205 
.67248 

43 
42 

8 
9 

8     f'7- 
9     6.4 

10     7.1 

20      14-3 

5-6 

6.4 

7-i 
14.1 

10 

ii 

9;4^75 

29 
30 
20 

9.64729 
.64772 

42 
43 

9.50533 
.  50562 

29 
29 

9.67296 
.67333 

42 
4? 

10 

ii 

30      21.5 
40      28.g 

50    35-8 

21  .2 
28.§ 

35-4 

12 

.48835 
.48864 
.48894 

29 

29 
29 

2;; 

.64815 
.64858 
.64901 

43 
42 
43 

.50591 
.50619 

•  50648 

29 
28 
29 

-67375 
.67418 
.67466 

.  4*  4*4 
»  tO)  tO)  t 

12 
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15 

9-48923 

9 

9.64943;  "f 

9.50677 

29 

9.67503 

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15 

16 

.48953 

3° 

20 

.649861  T^ 

.  50706 

29 

.67546 

43 

16 

4* 

17 

18 

.48983 
.49012 
.49042 

29 
29 
29 

.65029 
.65072 
.65114 

43 
42 

.50735 
.  50764 

•  50793 

28 
29 
29 

•67588 
•67631 
.67673 

.4*4*4 

»  tO)  IO)  t 

17 
18 

19 

6 
7 
8 

9 

4.2 
4-9 
5-6 
6-3 

20 

9.49071 

29 

9-6515?  ^ 

9.50821 

28 

9.67716 

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20 

20    14.0 

21 
22 
23 
24 

.49101 
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.49160 
.49189 

29 
29 

.65200 
.65243 
.65285 
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43 
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.50856 

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.50908 

•  50937 

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•67758 
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.67843 
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42 
42 
42 

21 

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30      21.0 
40      28.0 

50    35  .0 

25 
26 

27 
28 

9.49219  26 

.49248:  3 
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.49307   2~ 

9-65371  i  ?: 
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•65456i  I, 
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9.50965 
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9.67923 
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42 

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25 
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6     3.0 

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9.51109 

29 

9.68141 

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9     4-5 

4-4 

32 

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29 

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29 

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42 

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10     5.0 

20      10.  0 
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10 
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29 
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9.66651 
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42 

9.51823;  ii 
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9.69202 
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42 
42 

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55 
56 

30      14.0 

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57 

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29 
20 

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43 

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59 

60 

9.50243 

9.66864 

9.51965 

9.69414 

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Loe.  Vers.   Z> 

Loe.  Kxsec.   J> 

Log.  Vers.   /> 

Los*  Kxsec.  J> 

P.  P. 

417 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

48°  49° 


/ 

Log.  Vers. 

j> 

Log.  Exsec. 

D 

\  Log.  Vers. 

z> 

Lou.  Exsec. 

D 

> 

P.  P. 

0 

2 

3 
4 

9.51965 

.51994 
.  52022 
.  52056 
.52079 

28 
28 

28 

28 

9.69414 
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42 

42 
42 

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.  53676 
.  53704 
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2? 
28 
2? 

98 

9.71954 

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.72081 
.72123 

42 

42 
42 

0 

i 

2 

3 
4 

6 

8 
9 

9.5210? 

.52135 
.52164 
.52192 

.52220 

28 
28 

28 

28 

28 

-^0 

9.69625 
.69669 
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.69753 
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42 

42 
42 
42 
42 
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9.53787 
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.  53870 
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2? 

28 
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9*9 

9.72165 
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42 
42 
42 
42 

42 

6 

7 
8 

9 

10 

ii 

12 

13 

14 

9-52249 
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28 
28 

28 

28 

28 
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9.69838 
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42 
42 
42 
42 
42 

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9-53925 
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28 
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9.72376 
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42 
42 
42 
42 
42 

10 

ii 

12 
13 
14 

42 

42 

15 
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17 
18 

19 

9.52390 
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28 
28 

28 

9.70056 
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42 
42 
42 
42 

42 

9-54063 
•  54096 
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27 

2? 
27 

2? 

9.7258? 
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42 
42 
42 
42 
42 

15 

16 

17 
18 

19 

6     4-2 
7     4-9 
8     5-6 
9     6.4 
10     7.1 
20    14.1 

30      21.2 

4-2 
4.9 

I:! 

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14.0 

21.0 

20 

21 
22 
23 
24 

9.52531 
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28 

28 

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9.70262 
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•  70347 
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42 

42 
42 

.  s 

9.54206 

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27 
2? 

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9.72799 
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42 
42 
42 
42 
42 

20 

21 

22 

23 
24 

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50    35-4 

35-0 

3 

27 
28 
29 

9.52671 
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28 
28 
28 

28 

9.70474 
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•  70643 

42 

42 

42 
42 

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9.54338 
•  54365 
•  54393 
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.  54448 

27 

2? 
27 

2? 
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9.73010 
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42 
42 
42 
42 
42 
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25 
26 
27 
28 
29 

28 

6       2.§ 

7     3-3 
8     3-8 
9     4-3 

28 

2.8 

4.2 

30 

32 
33 
34 

9.52812 
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.  52868 
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28 
28 
28 
28 

9.70685 
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.  70770 
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42 
42 
42 
42 

42 
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9-54475 
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27 
27 
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9.73221 
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42 
42 
42 
42 
42 

30 

32 
33 
34 

10     4.7 
20     9.5 
30    14.2 
40    19.0 

So    23  .  7 

4-1 

9-3 
14.0 

23-3 

35 
36 

11 

39 

9.52952 
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28 
28 
28 
28 

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9.70897 

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42 
42 
42 
42 
42 

9.54612 

.  54639 
•  54667 
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.  54721 

27 

2? 
27 
2? 

9.73431 
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42 
42 
42 
42 
42 

35 
36 

39 

2? 

27 

40 

42 
43 
44 

9.53092 
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2o 
28 
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28 
28 
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9.7II08 
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42 
42 
42 
42 
42 

9-54748 
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•  54803 
•  54836 
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27 
2? 

2? 

2? 

9.73642 
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42 
42 
42 
42 
42 

40 

42 
43 
44 

2.7 
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20       9.1 
30      13-7 
40      l8.§ 

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3-6 
4.0 
4-5 
9.0 
13-5 
18.0 

45 
46 
47 
48 

49 

9.53231 
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2? 
28 
28 
28 

9.71320 
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42 
42 
42 
42 
42 

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9.54885 
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27 
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27 

9.73853 
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42 

42 

42 
42 

47 
48 
49 

50      22.9 

22.5 

50 

52 
53 
54 

9.53370 
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2*7 
28 
28 
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9.7I53I 
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42 
42 
42 
42 
42 

9.55021 
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27 
2? 

2? 

27 

9.74064 
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•74233 

42 
42 

42 
42 

50 

52 
53 

54 

11 

57 
59 

9.53509 
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28 

2? 
28 
2? 
28 

9-71743 
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42 
42 
42 
42 

Af 

9.55I57 
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27 

27 

27 

27 

27 

•7431? 
•74359 
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•74444 

42 

42 
42 
42 

42 

M 

57 
58 
59 

60 

9.53648 

9.71954 

9-55292 

9.74486 

00 

| 

Log.  Vers.' 

D 

og.  Exsec. 

D  t 

Log.  Vers.i 

J> 

oir.  Exseo. 

i> 

- 

P.  P. 

418 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

50°  51° 


; 

Loir.  Vers. 

D 

Loir.  Exsec. 

D 

Lo«r.  Vers. 

D 

Lojr.  Exsec. 

D 

p.  P. 

0 

9.55292 
.55319 

27 

O/? 

9.74486 
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42 

9.56900 
•  56926 

26 

72 

9.77012 
.77055 

42 

0 

I 

2 

3 

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*/ 

27 

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42 

.56953 
.  56979 

26 
26 

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4^ 

42 

2 

3 

4 

•55401 

27 

.74654 

42 

.57005 

„? 

.77181 

42 

4 

5 

9.55428 

27 

9.74696 

42 

9.57032 

25 

9-77223 

42 

5 

6 

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4 

-  57058 

25 

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ja 

6 

7 

.55482 

27 

.74781 

42 

.57085 

25 

27 

.77307 

42 

7 

8 
9 

.55509 
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27 

27 

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42 
42 

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5 

26 

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42 
42 

8 
9 

44 

6    .4.2 

42 

4.2 

10 

ii 

12 

13 

9-55563 
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27 

27 
27 

27 

9.74907 
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42 
42 
42 
42 

9.57164 
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26 
26 
26 

9-77433 
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42 
42 
42 
42 

10 

ii 

12 
13 

8     5-6 
9     6-4 
10     7.1 
20    14-1 

30      21.2 

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7.0 
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27 

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42 

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42 

U 

40      28.§ 

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15 

9.55698 

27 

9.75118'  ^ 

9.57296 

6 

9.77644 

42 

15 

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z 

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i+* 

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26 
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42 

16 

17 
18 

19 

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27 
27 

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2 

42 
42 

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26 
26 

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42 
42 

17 
18 

19 

20 

21 

9-55832; 

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27 

27 

22 

9-75328 
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42 
42 

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26 

26 

9-77854 
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42 
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A.2 

20 

21 

6     2.7 

27 

22 

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ti 

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22 

7     3-2 

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23 

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4^ 

23 

8     3-6 

3-6 

24 

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24 

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25 
26 
27 
28 
29 

9-55965 

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27 

27 

26 

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9-75539 
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42 
42 
42 
42 
42 

9-57559 
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26 
26 

26 

26 

26 

9.78064 
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42 
42 
42 
42 
42 

25 
26 
27 
28 
29 

20       Q.I 
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40      l8.3 
50      22.9 

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30 

9.56101 

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9.75750 

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9.57690 

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9.78275 

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30 

32 
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42 
42 
42 

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26 

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42 
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32 
33 

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26 

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36 
37 
38 
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10    i'i 

20     8.5 

30    13-2 
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21.6 

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9.56368 

9.76171 

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9.78696 

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9.56501 
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9.58082 
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9.58212 
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27 

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42 

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53 

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9.56767 

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26 

1 

26 

26 

9.76802 
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42 

42 
42 
42 
42 

9.58342 
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25 
26 
26 
26 
26 

9-79327 
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42 
42 
42 
42 
42 
42 

59 

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9.56900 

9.77012 

9.58471 

9-79537 

60 

Lou.  Vers.  I> 

Loe.  Exsec.  l  J> 

Loir.  Vers.   Z> 

Loer.  Exsec.  D 

' 

P.  P. 

4ig 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL   SECANTS. 

52°  53° 


/ 

Log.  Vers. 

D 

Log.  Exsec.   Z> 

Log.  Vers. 

j> 

Log.  Exsec. 

D 

/ 

p.  P. 

0 

I 

9.58471 
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26 

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9-79537 

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9.82062 
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42 

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3 

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42 

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25 

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4 

6 

9.58601 

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9-7974? 
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42 

9.60135 
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25 

25 

9.82272 
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42 
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6 

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26 

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42 

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25 

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7 
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9.58730 

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9-6038? 
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.60563 
.60589 

25 

25 

.82988 
•83031 

42 
42 

22 

23 

50    35-4 

35-o 

24 

.59090 

0» 

.80547 

42 

.60614 

25 

•83073 

42 

24 

3 

27 

9.59II6 

.59HI 
.5916? 

25 
25 
26 

9.80589 
.80631 
.80673 

42 
42 
42 

9  .  60639 
.60664 
.60689 

25 

H 

9.83II5 
.83157 
.83199 

42 
42 
42 

25 
26 

27 

28 

.59193 

2? 

.80715 

2 

.60714 

25 

.83241 

42 

28 

26 

23 

29 

•59218 

•8075? 

42 

.60739 

25 

•83283 

42 

29 

6     2.6 

2-5 

30 

32 

9.59244 
.59270 
•59295 

25 

9-80799 
.80841 
.80883 

42 
42 
42 

9.60764 
.60789 
.60814 

25 
25 
25 

9.83325 
.83368 
.83410 

42 
42 
42 

30 

32 

8     3-4 
9     3-9 
10     4.3 
20     8.  g 

3.4 

3-8 

4-2 

8-5 

33 
34 

.59321 
.59346 

25 

->s 

.80925 
.80968 

42 

42 

.60839 
.60864 

25 

25 

.83452 
.  83494 

42 

42 

33 
34 

30    13-0 

4°    17  3 

50      21.6 

12.7 
17.0 

21.2 

36 

9-59372 

•  5939? 

25 
25 

9.81010 
.81052 

42 
42 

9.60889 
.60914 

25 

25 

9-83536 

.83578 

42 

42 

35 
36 

37 
38 
39 

.59423 
•59449 
.  59474 

25 
25 

.81094 
.81136 
.81178 

42 
42 
42 

.60939 
.60964 
.60989 

25 

25 

.  83626 
.83663 
.83705 

42 
42 
42 

37 
38 
39 

40 

41 

9.59500 

25 

~P 

9.81226 
.81262 

42 
42 

9.61014 
.61039 

25 
25 

9.83747 
.83789 

42 

42 

40 

25 

6     2.5 

24 

,  2-4 

42 
43 

44 

.59551 
•59576 
.  59602 

25 

25 

25 
_  P 

.81304 
.81346 
.81388 

42 
42 
42 

.61064 
.61089 
.61114 

25 

25 

25 

'.83873 
.83916 

42 
42 
42 

42 
43 
44 

7     2.9 
•  8     3.3 
9     3-7 
10     4.1 

2  § 
3-2 
3-7 
4.1 
8  i 

45 
46 
47 

.59653 
•59678 

25 
25 

25 

9.81436 

.8i473 
.81515 

42 
42 
42 

9.61139 
.61164 
.61189 

25 
25 
25 

9.83958 
.84000 
.  84042 

42 
42 
42 

45 
46 

47 

30    12.5 
40    x6.g 
50    2o.g 

12.2 
.I6.§ 
20.4 

48 

.59704 

25 

.81557 

42 

.61214 

24 

.  84084 

42 

48 

49 

.59729 

25 

0? 

.81599 

42 

.61239 

25 

.84125 

42 

49 

50 

52 
53 

9-59754 
.5978o 

.59805 
.59831 

25 

25 
25 

25 

9.81641 
.81683 
.81725 
.8176? 

42 

42 
42 
42 

9.61264 
.61289 
.61313 
•61338 

25 

l\ 

25 

9.84168 
.84211 
.84253 
.84295 

42 
42 
42 
42 

50 

52 
53 

54 

.59856 

25 

.81809 

42 

.61363 

25 

.8433? 

42 

54 

55 

9.5988! 

25 

?p 

9.81851 

42 

9.61388 

25 

9.84379 

42 

AK 

55 

56 

.  59907 

25 

.81894 

42 

•61413 

24 

.84422 

42 

S6 

57 

•  59932 

0? 

.81936 

42 

.61438 

25 

.  84464 

42 

57 

58 

.59958 

25 

.81978 

42 

.61462 

24 

.  84506 

42 
AO 

58 

59 

•  59983 

25 

2? 

.  82020 

42 

42 

.6148? 

25 

•  84548 

42 

59 

60 

9.60003 

9  .  82062 

9.61512 

9.84596 

00 

Log.  \ers.   J>  Log.  Exsec.  i  7> 

Log.  Vers. 

z> 

Log.  Exsec. 

J> 

/ 

P.  P. 

420 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL    SECANTS. 

54°  55° 


Log.  Vers.!  D 

Los.  Exsec.  X> 

Log.  Vers. 

MM.  Kxsec. 

p.  P. 

0 

9.61512 

9-84590   ,-„ 

9.62984 

*>A 

9.87125 

0 

I 

2 

.61537 
.61562 

^4 

25 

.84632 
.84675 

** 

42. 

•63003 
.63032 

-4 
24 

.8716? 
.87209 

- 
42 
/i3 

2 

3 

.61585 

24 

.84717 

42 

.63057 

^4 

.87252 

42 

3 

4 

.6l6lT 

2> 

0  A 

.84759 

4.^ 
.a 

.63081 

z4 

i! 

.87294 

2 
.ft 

4 

5 

9.61636 

24 

9.8480! 

42 

/1  2 

9.63105 

24 

9-87336 

42 

5 

6 

7 

.6l66l 
.61685 

25 
24 

.84843 
.84886 

4^ 

42 

•63129 
.63154 

*4 

24 

.87379 
.87421 

4^ 

42 

/i5 

6 

7 

8 

.61716 

.  84928 

42 

.63178 

24 

.87463 

4.4 

A? 

8 

9 

.61735 

24 

.  84970 

4Z 
.a 

.63202 

*4 

.87506 

4* 

9 

10 

ii 

12 

9.61760 
.61784 
.61809 

25 

24 

24 

9.85012 
.85054 
.85097 

42 
42 
42 

9.63226 
•63256 
.63274 

24 
24 
24 

?A 

9.87548 
.87590 
.87633 

42 
42 

42 

10 

ii 

12 

13 

.61834 

27 

.85139 

42 

.63299 

24 

.87675 

2 

13 

.6l858 

.85181 

•+* 

.a 

•63323 

•*4 

.87717 

4^ 
.ft 

14 

42   42 

15 

16 

9.61883 
.61908 

24 

9.85223 
.85265 

42 
42 

A? 

9.63347 
.63371 

24 
24 

9.87760 
.87802 

42 
42 

\l 

6     4.2     4.2 
7     4-9     4-9 
8     5-6     5-6 

17 
18 

.61932 
.61957 

24 

24 

.85308 
.85350 

42 
42 

•6339! 
.63419 

24 
24 

.87844 
.87887 

42 
42 

A3 

17 

18 

9     6-4     6.3 
10     7.1     7.0 
20    14.1    14.0 

!  r9 

.61982 

3 

.85392 

4Z 

•63443 

^4 

1A 

.87929 

4^ 

19 

30      21.2      21.0 

j  20 

21 

22 
23 

9  .  62005 
.62031 
.62055 
.  62086 

24 
24 

24 
% 

9.85434 

.85476 
.85519 

-85561 

42 

42 
42 
/i5 

9.63468 
.63492 
.63516 
.63540 

24 
24 
24 
24 

9.87971 
.88014 

.88055 

.  88099 

42 
42 

42 
42 

20 

21 

22 
23 

50      35-4      35.0 

24 

.62105 

^4 

.85603 

4-4 

.63564 

^4 

.88141 

4^ 

24 

25 
26 

27 

9.62129 
.62154 
.62173 

24 
24 
24 

9.85645 
.85688 
•85730 

42 
42 
42 
,9 

9.63588 
.63612 
.63636 

24 

24 
24 
/7/T 

9.88183 
.88226 
.88268 

42 
42 

42 

11 

27 

25      24 

6     2.5     2.4 

28 

29 

.62203 
.6222? 

24 

24 

X,  J 

.85772 
.85814 

4^ 

42 
.a 

.63666 
.63684 

24 
24 

.88316 

.88353 

42 

42 

.a 

28 
29 

8     l'\     I'.l 
9     3-7     3-7 

30 

32 
33 
34 

9.62252 
.62275 
.62301 
.62325 
.62350 

24 
24 

24 
24 

24 

9.85857 
.85899 
.8594? 
•85983 
.86026 

42 
42 
42 
42 
42 

9.63703 
.63732 

.63756 
.63786 
.63804 

24 
24 

24 
24 
24 

9.88395 
.88438 
.  88486 
.88522 
.88565 

42 

42 
42 
42 
42 

30 

32 
33 

34 

10     4.1     4.1 

20       8.§       8.1 
30      I*-5      12.2 

40    16.5    16.3 
50    20.  §    20.4 

35 

9.62374 

9.86068 

42 

A? 

9.63823 

24 

1A 

9.8860? 

42 

43 

35 

36 

37 

.62399 
.62423 

24 

21 

.86116 
.86152 

42 
42 

.63852 
.63876 

*4 

24 

.88650 
.88692 

2 
42 

37 

38 

39 

.62448 
.62472 

24 
24 

.86195 
.86237 

42 

A  3 

.63900 
•63924 

24 
24 

.88734 
.8-8777 

42 

42 

A  3 

38 
39 

24     2§ 

j  40 

4i 
42 
43 
44 

9.62497 
.62521 
.62546 
.62576 
.62594 

24 
24 
24 

24 
24 

_  c- 

9.86279 
.86321 
.86364 
.  86406 
.86448 

42 
42 
42 
42 
42 

9.63948 
.63972 
.63996 
.64019 
.64043 

24 
24 
24 
23 
24 

9.88819 
.88862 
.88904 
.88947 

.88989 

42 
42 
42 
42 

42 

40 

42 
43 
44 

7     2.8     2.7 
8     3.2     3.1 
9     3-6     3-5 
10     4.0     3.9 
20     8.0     7.5 

30      12.0      II.7 

40    16.0    15.5 

45 

9.62619  :t 

9  .  86496 

42 

.ft 

9.64067  :? 

9.89031 

42 
Af) 

4S 

50      20.0      19.6 

46 
47 
48 

-62643 
.62668 
.62692 

f-Of 

24 
24 

•86533 
.86575 
.8661? 

42 
42 

42 

.  64091 
.64115 
.64139 

•L* 
24 
23 

.89074 

.89115 
.89159 

42 
42 

42 

46 
47 
48 

49 

.62715 

24 

_  T 

.86659 

42 

A  ft. 

.64163 

24 

.  89201 

42 

49 

50 

9.62741 

24 

9.86702 

42 
.ft 

9.641871  3 

9.89244 

A  0 
42 

50 

52 
53 

.62765 
.62789 
.62814 

24 
24 
24 

.86744 
•86785 
.86829 

42 
42 

42 

.64210 
.64234 
.64258 

*3 
24 
24 

.89285 
.89329 
.89371 

42 
42 

42 

52 
53 

54 

.62838 

24 

.86871 

42 

.  ~ 

.64282 

23 

.89414 

42 

54 

55 
56 
57 
58 

9.62862 
.62887 
.62911 
.62935 

24 

24 
24 

24 
-7/r 

9-86913 
.86956 
.86998 
.  87046 

42 
42 
42 

42 

9.64306 
.64330 
.64353 
.6437? 

24 
24 

23 
24 

9.89456 
.89499 
.89541 
.89583 

42 
42 
42 
42 

55 
56 

59 

.62960 

24 

24. 

.87082 

42 

4.2 

.64401 

23 

^A 

.89626 

42 

4.2 

59 

60 

9.62984 

9.87125 

9.64425 

9.89668 

60 

L',.. 

Loc.  Vers-i  /> 

Los.  Kxsec. 

n 

Los.  Vers.   D 

Los.  Kxsec.  J> 

' 

p.  p. 

421 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

56°  57 


/ 

Log.  Vers. 

Lop.  Exsec 

D 

Log.  Vers. 

D 

Log.  Exsec. 

D 

; 

P.  P. 

0 

I 

2 

9.64425 

•64448 
.64472 

23 

24 

9.89668 
.89711 
.89753 

42 
42 

A? 

^65859 
.65882 

23 
23 

9.92224 
.9226? 
.92310 

43 
42 

0 

i 

2 

3 

4 

.64496 
.64520 

24 

-2 

.89796 
.89838 

42 

42 

.  ~ 

.65905 
.65928 

23 
<->3 

.92353 
.92395 

43 

42 

3 

4 

5 

9-64543 

23 

9.89881 

42 

9.65952 

23 

9.92438 

43 

5 

6 

.6456? 

24 

.89923 

42 

.65975 

23 

.92481 

42 

6 

8 

.64591 
.64614 

23 

.  89966 
.90008 

42 
4? 

.65998 
.66021 

23 
23 

.92524 
.92565 

43 

42 

'8 

9 

.64638 

24 

.90051 

42 

.  66044 

23 

.  92609 

43 

9 

10 

ii 

12 

9  .  64662 
.64685 
.64709 

23 
23 
24 

9.90094 

•90136 
.90179 

43 

42 

42 

9.66068 
.66091 
.66114 

23 
23 

9.92652 
.92695 
.9273? 

42 
43 
42 

10 

ii 

12 

13' 
H 

.64733 
•64756 

23 
23 

.90221 
.90264 

42 

42 

-6613? 
.66166 

23 
23 

.92786 
.92823 

43 

42 

13 

43 

4? 

15 

9.64786 

24 

9-90305 

42 

9.66183 

23 

2s 

9.92866 

43 

10 

I   5-°- 

4.9 

16 

17 
18 

19 

.  64804 
.6482? 
.64851 
.64875 

23 

23 
24 

->3 

.90349 
.90391 

•  9°434 
.90475 

42 
42 
42 
.  ^ 

.66207 
.66230 
.66253 
.66276 

23 
23 
23 

.92909 

•92951 
.92994 

.93037 

43 

42 

43 
42 

16 

17 
18 

19 

8     57 
9     6.4 
10     7.1 

20      I4.§ 
30      21.5 

5-6 
6.4 

7-1 
14.1 

21.2 
28.§ 

20 

9.64898 

23 

9.90519 

42 

9.66299 

00 

23 

9  .  93080 

43 

20 

50    35-8 

35-4 

21 

.64922 

2l 

.90561 

42 

.66322 

23 

•93123 

43 

21 

22 

23 

24 

.64945 
.64969 
.64992 

23 
23 

.90604 
.90647 
.90689 

43 

42 
42 
.  ~ 

.66345 
.66363 
..6639! 

23 

23 

23 

•93165 
.93208 
•93251 

43 

43 
.  c 

22 

23 
24 

3 

27 
28 
29 

9.65015 
.65040 
.65063 
.65087 
.651  16 

24 
23 
23 
23 
23 

9.90732 

.90774 
.90817 
.  90860 
.  90902 

42 
42 
42 

43 
42 

9.66415 
.66438 
.  6646  i 
.66484 
.66507 

23 
23 
23 
23 

•93337 
•9338o 
.93422 
•93465 

42 
43 
43 
42 

43 

11 

27 
28 
29 

24 

6     2.4 
7     2.8 

*\ 

2.7 

30 

32 
33 

34 

9-65134 
.6515? 
.65181 
.65204 
.65228 

23 
23 
23 
23 

23 

9.90945 
.90987 
.91030 
.91073 
.91115 

42 

42 
42 

43 
42 
.  s 

9.66530 

•66553 
.66576 
.66599 
.66622 

-  23 
23 
23 
23 

9-93508 
•93551 
•93594 
•93637 
.  93680 

43 
42 

43 
43 

43 
.  ~ 

30 

31 

32 
33 
34 

8     3.2 
9     3-6 
10     40 

20       8.0 
30      12.0 

40    16.0 

50      20.0 

3-1 
3-5 
3-3 

11.7 

15-6 
19.6 

35 
36 
37 
38 

9.6525! 
.65275 

•65298 
.6532! 

23 
23 
23 
23 

9.91158 
.91206 
.91243 
.91286 

42 
42 
42 
43 

9.66645 
.66668 
.66691 
.66714 

23 
23 
23 

9.93722 
.93765 
.93803 
.93851 

42 
43 
43 
43 

35 
36 
37 

39 

.65345 

23 

~z 

.91328 

42 

.66737 

23 

•93894 

42 

39 

40 

9.65368 

23 

9.91371 

42 

9.66760 

23 

9-93937 

43 

40 

42 
43 
44 

•65392 
.65415 

.65439 
.65462 

23 
23 
23 
23 

.91414 
•9H56 
.9H99 
.91541 

43 

42 

42 
42 

.66783 
.66805 
.66828 
.6685! 

23 

23 

23 
23 

.9398o 
-94023 
.  94066 
.94109 

43 
43 
43 
43 

.  ~ 

42 
43 
44 

23 

6     2.3 
9     3'4 

22 

2.2 
2.6 

3-4 

45 
46 
47 
48 

49 

9.65485 
.65509 
.65532 
•65556 
.65579 

23 
23 
23 

23 

9.91584 
.91627 
.91669 
.91712 
.91755 

42 
42 

43 
42 

9.66874 
.6689? 
.66926 
•  66943 
.66966 

23 

23 

22 
23 

9.94151 
.94194 
.9423? 
.  94286 

.94323 

42 

43 
43 
43 

43 

45 

46 
47 
48 
49 

10     3<§ 

20       7.6 
30      II-5 

40     15.  | 
50     19.1 

3-7 

,11 

50 

9.65602 
.65626 

23 
23 

9-9*79? 
.91846 

42 
43 

9.66989 
.67012 

23 

9-94366 
.  94409 

43 
43 

50 
51 

52 

.65649 

23 

?7 

.91883 

2 

•67034 

22 

.94452 

43 

S2 

53 

54 

.65672 
.65696 

23 
23 

.91926 
.91968 

42 

•6705? 
.  67086 

23 

23 

.94495 
•94538 

43 
43 

53 
54 

55 
56 
57 
58 
59 

9.65719 
.65742 
.65765 
.65789 
.65812 

23 
23 
23 
23 
23 
27 

9.92011 
.92054 
.92095 
.92139 
.92182 

42 

43 
42 

43 

42 

A2. 

9.67103 
.67126 
.67149 

.67171 
.67194 

22 
23 

23 

22 

23 

9.94581 
.94624 
.9466? 
.94716 
•94753 

43 
43 
43 
43 
43 

55 
56 
57 
58 
59 

60 

9-65835 

9.92224 

4^ 

9.67217 

9  .  94796 

43 

60 

Log.  Vers.   D   ^g.  Exsec.  D 

Log.  Vers.   D 

.HIT-  Exsec. 

7> 

f 

P.  P. 

422 


TABLE    VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

58°  59° 


Los:.  Vers. 

n 

M$.  Kxsec. 

J> 

Log.  Vers. 

Z> 

MS.  Kxsec. 

D 

p.  P. 

0 

2 

3 
4 

9.67217 
.67240 
.67263 
.67285 
-67308 

23 
23 

22 
23 

_2 

9-94796 
•  94839 
.94882 

.94925 
.94968 

43 
43 
43 
43 

9.68571 
.68593 
"  .68615 
.6863? 
.68660 

22 

22 
22 
22 

•9743° 
•97473 
.97517 
.97566 

43 
43 
43 
43 

0 

I 

2 

3 
4 

6 

8 
9 

9.67331 
-67354 

•67376 
.67399 
.67422 

23 

22 

23 

22 

9.95011 
.95054 
.9509? 
.95140 
.95183 

43 
43 
43 
43 
43 

9.68682 
.68704 
.68727 
-68749 
.6877? 

22 
22 
22 
22 
22 

9.97603 

.97647 
.97696 

•97734 
.9777? 

43 
43 
43 

21 

6 

7 
8 

9 

44    43 

6     4.4     4-3 

10 

ii 

12 

13 

9.67445 
.67467 
-67490 
-67513 
.6753$ 

22 
22 

23 
22 
2  a 

9.95226 
.95269 

.95313 
•95356 
-95399 

43 
43 
43 
43 
43 

9.68793 
.68816 
.68838 

.68866 
.68882 

22 

22 
22 
22 
22 
_2 

9.97826 
.97864 
•9790? 
•97951 
•97994 

3 

43 
43 
43 

10 

ii 

12 
13 
14 

7     5-1     5-i 
8     5-8     5-8 

10     7.3     7.2 

20      I4-6      M.S. 
30      22.  0      21.7 
40      29.3      29.0 

15 
i6 

17 
18 

9.67558 
.67581 
.67603 
.67626 
.67649 

2 

23 

22 
22 

23 

9.95442 
•95485 
•95528 

-95571 
.95614 

43 
43 
43 
43 
43 

9.68905 
.68927 
.  68949 
.68971 
-68993 

22 
22 
22 
-22 
22 

9-98038 
.98081 
.98125 
.98168 
.98211 

43 
43 
43 
43 
43 

15 

16 

17 

18 
19 

50    36>  6    36-2 

20 

21 

22 
23 
24 

9.67671 
•67694 
.67717 

-67739 
.67762 

22 

23 
22 
22 

,,2 

9.95657 
.95706 

•95744 
.95787 
-95830 

43 
43 
43 
43 
43 

9.69016 
.69038 
.  69060 
.69082 
.69104 

22 
22 

22 
22 
22 

9.98255 
.98293 
.98342 
.98385 
.98429 

43 
43 
43 
43 

43 

A  5 

20 

21 

22 

23 
24 

.   fa 
1   |1 

9     64 
10     7.1 

3 

27 
28 
29 

9.67784 
.67807 
-67830 
.67852 
.67875 

22 

23 

22 
22 

•95916 
•95959 
.96002 
.  96046 

43 
43 

43 

43 
43 

9.69125 
.69149 
.69171 
.69193 
.69215 

22 

2 

2 

2 

9-98472 
,98516 

•98559 
.  98603 

.98647 

43 
43 
43 
44 
43 

25 
26 

27 
28 

29 

20    14  3 

30   21.5 
40   28-6 

50    35-8 

32 
33 
34 

9-67897 
.67920 

.67942 
.67965 
-67987 

^2 

22 
22 
22 
22 
_s 

9  .  96089 
.96132 
.96175 
.96213 
.96261 

43 
43 
43 
43 
43 

2 

9.69237 
.69259 
.69281 
-69303 
.69325 

22 
22 
22 
22 

9.98696 
.98734 
-9877? 
.98821 
.98864 

43 
43 
43 
43 
43 

30 

32 

33 
34 

23     22 

6     2.3     2.2 

35 
36 

39 

9.68010 
.68032 
.68055 
.68077 
.68100 

22 
22 
22 
22 
22 
1% 

9.96305 
.96348 
.96391 
•96434 
.96478 

43 
43 
43 
43 

43 

9-69347 
.69369 
.69392 
.69414 
.69436 

22 
22 
22 
22 

22 

9.98908 
.98952 
.98995 
.99039 
.99082 

43 
44 
43 
43 
43 

P 

37 
38 
39 

7     2.7     2.6 
8     3.0     3.0 
9     3-4     3-4 
10     3.§     3-7 
20     7-6     7-5 

30      II-5      1  1.  2 
40      15.3      15.0 
50      ig.I      18.7 

I  40 

4i 

42 

43 
44 

9.68122 
.68145 
.68167 
.68190 
.68212 

22 
22 
_a 

" 

9.96521 
.96564 
.96607 
.96656 
.  96694 

43 
43 
43 
43 
43 

9.69458 
.69480 
.69502 
.69524 
.69546 

22 
22 
22 
22 

9.99126 
.99170 
.99213 

.99257 
.99306 

43 
44 
43 
43 
43 

40 

42 
43 
44 

45 
46 
47 
48 

I  49 

9.68235 
•68257 
.68280 
.68302 
.68324 

22 
22 
22 
Oa 

9.96737 
.96786 
.96824 
.96867 
.96916 

43 
43 
43 
43 
43 

9.69568 
-69590 
.69612 
.69634 
.69656 

22 
22 
22 
22 
22 

9-99344 
.99388 

•99431 
•99475 
•995'9 

44 
43 
43 
44 
43 

45 
46 
47 
48 
49 

22     21 

6       2.2       2.1 

9     33     3-2 
10     3.6     3.6 

50 

52 
53 
54 

9.68347 
.68369 
.68392 
.68414 
.68435 

22 

22 

22 
-2 

9.96953 
.96997 
•  97040 
•97083 
.97127 

43 
43 
43 
43 
43 

9.69678 
.69700 
.69721 
.69743 
.69765 

22 
22 
21 
22 

22 

9.99562 
.99605 
•99650 
.99694 
•9973? 

43 
44 
43 
44 
43 

50 

52 
53 
54 

20       7.3       7.1 

30    ii  o    10.7 
40    14-$    14-3 
50    18.3    17.9 

55 
56 

59 

9.68459 
.68481 
•68503 
.68526 
.68548 

22 
22 
22 
22 
2*> 

9.97170 

.97213 
.97257 
.97300 
•97343 

43 
43 
43 
43 
43 

*  5 

9.6978? 
.69809 
.69831 
.69853 
-69875 

22 
22 
22 
21 
22 
2° 

9.99781 
•99825 
.99863 
.99912 
9.99956 

44 
43 
43 
44 

43 

A  A 

55 
56 
57 
58 
59 

«0 

9.68571 

9-97387 

9.69897 

10.00000 

44 

60 

Loir.  \ers. 

1) 

jog.  Kxsec. 

/> 

Los.  Vers. 

T> 

Loir.  Kxsec. 

J> 

' 

P.  P. 

423 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL    SECANTS. 

6O°  61° 


/ 

Log.  Vers 

JD 

Los?.  Kxsec. 

J> 

Log.  Vers 

D 

Log.  Exsec. 

J> 

\ 

p. 

P. 

0 

I 

2 

3 

4 

9.69897 
.69919 
.69948 
.  69962 
.69984 

22 
21 

22 

22 

10.00000 
.00044 
.0008? 
.0013! 

.00175 

44 
43 
44 
43 

9.71197 
.71218 
,71239 
.71261 
.71282 

21 
21 
21 
21 
_  f 

10.02639 
.02684 
.02728 

.02772 
.02815 

44 
44 

44 

44 
.  -> 

0 

i 

2 

3 

4 

6 

8 
9 

9.70005 
.70028 
.70050 
.70072 
.70093 

22 
21 
22 
22 
21 

IO.OO2I9 
.  00262 
.00306 
.00350 
.00394 

44 
43 
44 
44 
43 

9.71304 
.71325 
•71346 
.71368 

.71389 

21 
21 
21 
21 
21 
of 

IO.O286I 
.02905 
.02949 
.02994 
.03038 

44 
44 
44 
44 
44 

A  T 

I 

7 
8 

9 

6    ^ 

5 
3 

44 

4.4 

10 

ii 

12 

13 
H 

9.70115 
.70137 
.70159 
.70181 
.  70202 

21 

22 
22 
21 

10.00438 
.00482 
.00525 
.00569 
.00613 

44 
44 
43 
44 
44 

9.7HI1 
•71432 
.71453 
.71475 
•7  H96 

21 
21 
21 
21 

10.03082 
.03127 
.03171 
.03215 
.03260 

44 

44 
44 
44 
44 

10 

ii 

12 

13 
14 

1     I 
9     6 
10     7 
20    15 

30      22 
40      30 

5°    37 

o 
7 
5 
o 

o 

5-2 

5-9 
6.7 
7-4 
14-jj 

22.2 
29-6 

37*  l 

II 

17 

18 
19 

9.70224 
.  70246 
.70268 
.70289 
-.7031! 

22 
21 
22 
21 

22 

'IT' 

10.00657 
.00701 
.00745 
.00789 
.00833 

44 
43 
44 
44 
44 

9-7151? 
.71539 
.71566 

.7158? 
.71603 

21 
21 
21 
21 
21 
_~ 

10.03304 

•03348 
.03393 
•0343? 
.03481 

44 
44 
44 
44 
44 

15 
16 

17 
18 

19 

20 

21 

22 

23 

24 

9-70333 
.70355 

•70376 
•70398 
.  70420 

21 

22 
21 
22 
21 
-->? 

10.00876 
.  00920 
.00964 

.01008 
.01052 

43 
44 
44 
44 
44 

9.71624 

.71645 
.71667 
.71688 
.71709 

21 
21 
21 
21 
21 

10.03526 
.03576 
.03615 
.03659 
•03704 

44 
44 
44 
44 
44 

20 

21 

22 

23 

24 

6    J 

I    I 

9     6 
10     7. 

4 

4 

i 

8 

8 
i 

43 

4-3 

5-i 
5-8 
6.5 

7-2 

25 

26 

27 
28 
29 

9.70441 
.70463 
.70485 
.70507 
.70523 

21 
22 
21 
22 
21 
_  ~ 

10.01095 
.01146 
.01184 

.01228 

.01272 

44 
44 
44 
44 
44 

9-71730 
.71752 
.71773 
.71794 
.71815 

21 
21 
21 

21 
21 
_  ^ 

10.03748 

.03793 
.03837 
.03881 
.03926 

44 
44 
44 
44 

44 
.  j 

% 

27 
28 
29 

20    14. 

30      22. 
40      29. 

50      36. 

6 

0 

1 

6 

14-5 
21.7 

S?Q.O 
36.2 

30 

3i 
32 
33 
34 

9.70550 
.70572 

.70593 
.70615 
•70635 

21 

22 
21 
21 
21 

10.01315 

.01366 
.01404 
.01448 
.01492 

44 
44 
44 
44 
44 

9.71837 
.71858 

.71879 
.71906 
.71922 

21 
21 
21 
21 
21 

10.03976 
.04015 
.04059 
.04104 
.04149 

44 
44 

44 
44 
45 

30 

3i 
32 
33 
34 

2 

6       2. 

2 

2 

21 

2.1 

3 

37 
38 
39 

9.70658 
.70680 
.70701 
.70723 
.70745 

22 
21 
21 
21 

22 

10.01535 
.01586 
.01624 
.01663 
.01712 

44 
44 
44 
44 
44 

9-7I943 
.71964 

.71985 
.72005 
.72028 

21 
21 
21 
21 
21 

10.04193 
.04238 
.04282 
.04327 
.04371 

44 
44 
44 
44 
44 

11 

37 
38 
39 

8        2. 

9     3- 
10     3- 
20    r  7  . 
30    xi. 
40    14. 
50    18. 

9 

a 

8 

0 

1 

2-5 

28 

3-2 
3.6 
7-1 
10.7 

•4-j 

17.9 

40 

41 
42 

43 

44 

9.70765 
.70788 
.  70809 
.70831 
.70852 

IT 

21 
21 
21 
21 
21 

10.01755 
.01806 
.01844 
.01889 
.01933 

44 
44 

44 

44 
44 

9-72049 
.72070 
.72091 
.72112 
•72133 

21 
21 
21 
21 
21 

10.04416 
.  0446  i 

.04505 
.04550 
.04594 

44 
45 
44 
44 
44 

40 

4i 

42 
43 
44 

45 
46 

47 
48 

49 

9.70874 
.70896 
.7091? 
.70939 
.70966 

21 
21 
21 
21 
_~ 

10.01977 

.02021 
.02065 
.O2IO§ 
.02153 

44 
44 
44 
44 
44 

9.72154 
.72176 
.72197 
.72218 
.72239 

21 
21 

21 
21 
21 
_c 

10.04639 
.  04684 
.04728 

•04773 
.04818 

45 
44 
44 
44 
45 

.  7 

45 
46 

47 
48 
49 

6 

7 
8 
9 

10 

21 

2.  I 

2.4 
2.8 

3-i 
3-5 

50 

51 

52 
53 

54 

9.70982 
.71003 
.71025 

.71045 
.71068 

21 
21 
21 
21 
21 
_o 

10.0219? 
.02242 
.02286 
.02330 

.02374 

44 
44 
44 
44 
44 

9.72266 
.72281 
.72302 
.72323 
.72344 

21 
21 
21 
21 
21 

10.04862 
.0490? 
.04952 
.04995 
.05041 

44 

45 
44 
44 

45 
.  f 

50 

5i 
52 

53 
54 

20 

30 
40 

50 

I 
I 
I 

7.0 

3-5 
^.0 

7-5 

55 
56 
57 
58 
59 

9.71089 
.71111 
.71132 
.7H54 
.71175 

21 
21 
21 
21 
21 
r>j 

10.02418 
.02463 
.02507 
.02551 
.0259§ 

44 

44 
44 
44 
44 

AA. 

9.72365 

.72385 
.72408 
.72429 
.7245° 

21 
21 
21 
21 
21 
2  1 

10.05086 
.05131 
.05175 
.05226 
.05265 

44 
45 
44 
45 

44 

A  C 

55 
56 
57 
58 
59 

60 

9.71197 

10.02639 

9.72471 

10.05310 

60 

' 

Log.  Vers. 

/> 

,»i.'.  Exsec. 

I) 

Loc.  Vers. 

Z> 

jojr.  Kxsec. 

u 

' 

P. 

424 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL    SECANTS. 

62°  63° 


/ 

Log.  Vers. 

Log.  Exsec.|  1> 

Log.  Vers. 

D 

josr.  Exsec. 

D 

P.  P. 

0 

I 

2 

9.72471 
.72492 
.72513 

21 
21 

10.05310 
•05354 
.05399 

44 

45 

9.73720 
.73740 
.73761 

20 
20 

<J  T 

I0.o8oi5 
.08061 
.08106 

45 
45 

0 

I 

2 

3 

.72534 

.05444 

45 

.73782 

«A 

.08151 

3 
,P 

3 

4 

.72555 

.05489 

44 

.73802 

.08197 

45 

4 

2 

46 

5 

9.72576 

21 

10.05534 

45 

9.73823 

20 

10.08242 

45 

5 

6     4-6 

4.6 

6 

8 
9 

.72597 
.72618 
.72639 
.72660 

21 
21 

21 

.05579 
.05623 

.05663 
.05713 

45 
44 
45 
45 

.73843 
.73864 
.73884 
.73905 

20 
20 

21 

.08288 
-08333 
.08379 
.08424 

45 
45 
45 
45 

A  ? 

6 

8 
9 

7      5-4 
8     62 
9     70 
10     7.7 

20      15.5 
30      23.2 

if 

7  <j 
'5-3 
23.0 

10 

ii 

9,72681 
.72701 

21 
20 

10.05758 
.05803 

45 
44 

9.73926 
•73946 

2O 
20 

10.08470 
.08515 

45 
45 

10 

n 

40      31.0 
5°      38.7 

38  3 

12 

.72722 

.05848 

45 

.73967 

^n 

.08561 

5 

12 

13 

H 

.72743 
.72764 

21 

.05893 
.05938 

45 

45 

.73987 
.74008 

20 

-£. 

.08605 
.08652 

45 

A  ? 

13 

H 

u 

!  72805 

21 

10.05983 
.06028 

45 

9-74028 

.74049 

20 

10.0869? 
.08743 

45 
46 

15 

16 

45 

45 

17 

.7282? 

.06072 

.7406§ 

.08789 

45 

4p 

17 

7     5-3 

5  2 

18 

'9 

.72848 
.72869 

21 

.0611? 
.06162 

45 
45 

.74090 
.74110 

20 

.08834 
.08880 

5 
45 

.{. 

18 
19 

8     66 
9     68 
10     76 

6.0 
6-7 
7-5 

20 

21 

22 

9.72890 
.72911 
.72931 

21 

20 

10.06207 
.06252 
.06297 

45 
45 
45 

.7415? 
.74172 

2O 
20 
20 
s>rt 

10.08926 
.08971 
.09017 

40 
45 
45 

20 

21 

22 

23      I5.I 
30      22.7 

40    3°-3 
50    37-9 

22  5 
30.0 
37  5 

23 

24 

.72952 
.72973 

21 

.06342 
.0638? 

45 
45 

.74192 
.74213 

20 

.09062 
.09103 

46 

.  ? 

23 
24 

25 

9.72994 

21 
2O 

10.06432 

45 

4.C 

9.74233 

2O 

20 

10.09154 

45 
46 

25 

27 
28 

29 

.73015 
.73036 
.73057 
.7307? 

21 
21 
2O 

.0647? 
.06522 
.06568 
.06613 

45 
45 
45 

.74254 
.74274 
.74294 

.74315 

20 
20 
20 

.09200 

.09245 
.09291 

•09337 

45 
45 
46 

A  ? 

27 
28 
29 

44 

6     4.4 
7     5.2 
8     5  9 

30 

9.73098 

10.06658 

45 

1  r 

9-74335 

2O 

10.09382 

45 

Aft 

30 

9 

6  7 

32 
33 
34 

-73119 

•  73HO 
.73161 
.73181 

20 
21 

20 

.06703 
.06748 
.06793 
.06833 

45 
45 
45 
45 

.74356 
.75376 
•74396 
.74417 

20 
20 
20 

.09423 
.09474 
.09520 
.09566 

4° 
46 

45 
46 

A  ? 

32 
33 

34 

20      14.  § 
30      22.2 
4<>      29-6 

50    37  i 

vr»VO  rv 
CO  CO  CO 

9.73202 
.73223 
.73244 

21 
21 

20 

10.06883 
.06923 
.06974 

45 
45 

A  C 

9-7443? 
.74458 
.74478 

2O 

20 
20 

10.09611 
.0965? 
.09703 

45 
46 
46 

35 
36 

37 

38 
39 

.73265 

20 

.07019 
.07064 

45 
45 

A  ? 

•74498 
.74519 

20 

.09749 
-09795 

46 

.£. 

38 
39 

21 

6       2  I 

20 

2.6 

40 

9-73306 

21 
20 

10.07109 

45 

4e 

9-74539 

2O 
2O 

10.09841 

40 

40 

7     2.4 
8     28 

2'4 

41 

.73327 

.07154 

4P 

•74559 

.09885 

ft 

41 

9     3-5 

3.1 

42 
43 
44 

•73348 
.73368 
.73389 

20 
21 
~s 

.07200 
.07245 
.07290 

5 
45 
45 

.  a 

.74580 
..74606 
.  74626 

20 
20 

.09932 
.09978 
.10024 

46 
46 

A  A 

42 
43 
44 

10     3-5 

20       7.0 
30      10.5 
40      14.0 

10.2 

'3-6 

45 

9-734io  S 

10.07335 

45 

9.74641 

2O 

10.  10076 

40 

4S 

46 

.73430 

.07386 

45 

A  8 

.74661 

20 

.10115 

Aft 

46 

47 
48 

.7345? 
.73472 

25 
2T 

.07426 
.07471 

45 

45 
A? 

.74681 
.74702 

20 

.  10162 
.  .  10208 

4° 

45 

Aft 

47 
48 

49 

•73493 

ort 

.07516 

45 

A  ? 

.74722 

.10254 

4U 

Aft 

49 

50 

51 

52 
53 
54 

9-735I3 
•73534 
'•73555 
•73575 
.73596 

20 
21 
20 
20 

10.07562 
.07607 
.07652 
.0769? 
-07743 

45 
45 
45 
45 
45 

9.74742 
.74762 
.74783 
•74803 
.74823 

20 
20 
20 
20 

10.10300 
.  10346 
.10392 
.  10438 
.  10484 

40 

46 
46 
46 

46 

A  A 

50 

5i 
52 
53 

54 

2O 

7     2.3 
8     2.6 
9     3-o 
3-3 

20       6.5 

P 

57 

9.73617 

.7363? 
.73658 

21 
20 
20 

10.0778^ 
.07834 
.07879 

45 
45 

45 

9.74844 
.74864 
.74884 

2O 

20 
20 

10.  10530 
•10575 
.  10622 

46 

46 
46 

Aft 

ii 

57 

40   133 

50    16.5 

.73679 

2O 

.07924 

5 

.74904 

.10663 

40 

Aft 

58 

59 

.73699 

26 

.07970 

5 

A? 

.74924 

.10714 

4U 

59 

60 

9.73720 

10.08015 

45 

9-74945 

10.  10766 

1 

60 

'    Log.  Vers. 

7> 

Log.  Exsec.   D   Log.  Vers.  D 

Log.  Exseo.   D 

P.  P.      1 

TABLE    VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

64°  65° 


/ 

Log.  Vers. 

D 

Log.  Exsec. 

J> 

Log.  Vers. 

J> 

jog.  Exsec. 

J> 

; 

p.  p. 

0 

9-74945 

IO.I0766 

/i2 

9.76145 

TO 

10.1355! 

0 

I 

2 

.74965 

20 

.  10807 
.10853 

46 
46 

.76166 
.76186 

19 

20 

.13598 
.13645 

47 
47 

2 

3 

4 

.75005 
.75026 

20 

.  10899 
.  10945 

46 

A  f\ 

.76206 
.76225 

19 

.13692 
.13739 

47 
47 

3 
4 

6 
8 

9.75046 
.75066 
.75086 
.75106 

2O 
20 
20 
2O 

10.10991 
.1103? 
.  11084 
.  1  1  1  30 

40 
46 

46 
46 

9.76245 
.76265 
.76285 
•  76304 

2O 

20 
19 

10.13785 

.13833 
.13886 

.1392? 

47 
47 
47 

47 

6 
8 

6     4.8 
7     5-6 
8     6.4 
9     7-2 
10     8.0 

I 

7-1 
7-9 

9 

•75125 

.11  176 

f- 

.76324 

•13974 

47 

9 

20    16.0 

15-8 

10 

9-75H7 

10.11222 

46 

A9 

9.76344 

2O 

10.1402! 

47 

10 

40    32.0 

ii 

.75^7 

.  1  1  269 

46 

.76364 

J9 

.  14063 

47 

ii 

50    40.0 

39-6 

12 

.75187 

.11315 

A? 

.76384 

.14115 

47 

12 

13 

H 

.75207 

•7522? 

20 

.II36T 
.1140? 

46 
46 

.76403 
.76423 

19 

20 

.  14162 
.14210 

4? 

13 

II 

17 

18 

9;  7526? 
.7528? 
.75308 

2O 
2O 
20 
20 

10.11454 
.11506 

.H546 
.H593 

46 
46 
46 

9-76443 
.76463 
.76482 
.76502 

20 
19 
19 

10.14257 
.14304 
.1435* 

•  H398 

47 
47 
47 
4? 

11 

17 

18 

47 

6     4.7 

46 

4-6 

?:j 

19 

.75328 

.11639 

.76522 

2O 

.U445 

47 

,c 

19 

9     7.0 

20 

9.75348 

10.11685 

46 
,.« 

9.76541 

19 

10.14493 

47 

20 

20    15-6 

15-5 

21 
22 
23 

'75388 
.75408 

2O 
20 

20 

.11732 

.11778 
.11825 

46 
46 

46 

.  ? 

.7656! 
.76581 
.  76606 

20 
5§ 

.  14540 
.14587 
.  H634 

47 
47 

4? 
AG 

21 

22 
23 

30    23.5 
40    31.  | 

50    39-i 

23.2 
31.0 

38.7 

24 

.75423 

.11871  ^o 

.  76626 

20 

.  14682 

47 

24 

25 
26 

27 

9-75448 
.75468 
.75488 

2O 
20 
20 

10.1191? 
.11964 

.12010 

40 

46 
46 

9  .  76640 
.76659 
.76679 

20 

10.14729 

.  H776 
.14823 

47 
4? 
47 

y 

27 

*6 

28 

.75508 

.12057 

46 

•  2 

.76699 

J§ 

.14871 

A*7 

28 

6 

4.6 

29 

.75528 

.12103 

46 

.76718 

19 

.14918 

47 

29 

7 
8 

i'i 

30 

9-75548 

IO.I2I50 

46 

9.76738 

20 

10.  14965 

47 

30 

9 

6.9 

31 

•75568 

.12196 

4§ 

.76758 

19 

i  A 

.15013 

K 

31 

TO       7<6 

32 
33 

34 

.75588 
.75608 
.75628 

20 
20 

.12243 
.12289 
.12336 

46 
46 

.7677? 
.76797 
.76817 

19 

20 

.  i  5066 
.15108 
.15155 

4? 

47 

.  c 

32 
33 
34 

30    23.0 
40    30.  g 
5°    38-3 

35 
36 

37 

9^7566§ 
.75688 

2O 
20 
20 

10.12383 
.I242§ 
.12476 

4V 
46 

46 
.  p 

9.76835 
.76856 

.76875 

19 
19 

10.  15202 

.15250 
.1529? 

4? 
4? 
4? 

11 

37 

38 
39 

•75708 
.75728 

20 

.12522 
.12569 

46 
46 

.76895 
.76915 

2O 
19 

.15345 
.15392 

4? 

.« 

38 
39 

20 

6     2.6 

20 

2.O 

40 

9-75748 

IO.I26l6 

4/ 

9.76934 

iy 

10.15440 

47 

40 

7     2.4 

2-3 

41 

.7576§ 

.12662 

46 

•  .76954 

I9 

.1548? 

7 

41 

8     2.7 

2-6 

42 
43 
44 

.7578§ 
.75803 
.75828 

20 

'9 

.12709 
.12756 
.12802 

46 
47 
46 

.76973 

.76993 
.77012 

i  r\ 

19 

:!||il 

.15630 

4? 
4? 

.  0 

42 
43 

44 

10     3.4 

20       6.§ 
30      10.2 

40   13-6 

1:1 

IO.O 

13.3 

45 
46 

47 
48 

49 

9.75848 
.75868 
.75888 
.75908 
.75928 

20 
20 
20 
20 

TO 

10.12849 
.12896 
.12942 
.  12989 
.13036 

46 
47 
46 
47 
46 

9-77032 
.77052 
.7707? 
.77091 
.77110 

20 
19 
19 
19 
19 

10.  15678 

.15725 

.15773 
.15826 

.15868 

48 
4? 
4? 
4? 
48 

.£ 

45 
46 
47 
48 
49 

50   17.1 

i6.g 

50 

9-75947 

19 

10.  13083 

47 

9.77130 

!5 

10.15916 

47 
A 

50 

19 

52 

.7596? 
.7598? 

20 

.13130 
.13176 

47 
46 

.77U9 
.77169 

19 

.15963 

.16011 

47 
48 

52 

6 

I 

1.9 

1.1 

53 

.7600? 

in 

.13223 

47 

.77188 

.16059 

4? 

A*7 

53 

9 

2.9 

54 

.76027 

19 

.13270 

40 

.77208 

.16105 

47 

.  o 

54 

20 

55 

56 

9.76047 
.76067 

20 

10.13317 
.13364 

47 
47 

9.7722? 
.77247 

jy 

10.16154 
.16202 

45 
4? 

56 

30 
40      I 
50      I 

9-7 
6.2 

57 
58 

.76087 
.76105 

19 

.13411 

•1345? 

47 
46 

.77265 
.77286 

19 

.16250 
.16298 

48 

H 

59 

.76125 

2O 

.13504 

47 

.77305 

I9 

.16345 

4.8 

59 

60 

9.76145 

10.13551 

47 

9.77325 

19 

io.  16393 

60 

Log.  Vers.   Z»  Log.  Exsec.  /> 

Log.  Vers.   j> 

Log.  Exsec.  />    '         P.  P. 

426 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL   SECANTS. 

66°  67° 


i 

Loir.  Vers. 

J> 

Log.  Exsec. 

Z> 

Log.  Vers. 

Z» 

Loe.  Exsec. 

D 

P.  P. 

0 

I 

2 

3 
4 

9-77325 

.77344 
.77363 
.77383 
.77402 

19 
19 
19 

i.9 

T  A 

.16441 
.  16489 
.16537 
.16585 

I 

9.78481 
.78500 
i  .78519 
.78538 
.78557 

19 
19 
19 

10.  19293 

.19342 

•I939I 
•19439 

49 
49 
48 
49 

0 

i 

2 

3 

4 

CO 

6 

8 
9 

9-77422 
.77441 
.77461 
.77480 

•77499 

J9 

If 

19 

19 

T  A 

10.  16633 

.16686 

.16775 
.16824 

I 

9-78576 
.78595 
.78614 
•78633 
.78652 

J9 
19 
19 
19 
19 

10.1953? 
•19586 
.19635 
.19684 

•19733 

49 
49 
49 
49 
49 

I 

7 
8 

9 

6 
7 
8 
9 

10 

20 
3° 

5° 

5-o 

ii 

7-5 

8-3 
16.6 
25.0 

4.9 
5-8 
6  6 
7-4 

8.2 

16.5 
24.7 

10 

ii 

12 
13 

U 

9.77519 
•77538 
•7755? 
.77577 
.77596 

*9 

19 
19 
19 
19 

IO.I6872 
.  16926 
.16963 

.17016 
.17064 

48 
48 

,0 

9.78671 
.78696 
.78709 
.78728 
.7874? 

19 

19 
19 

10.19782 
.19831 
.19886 
.19929 
.19979 

49 
49 
49 
49 
49 

10 

ii 

12 
13 

40 
50 

33  3 

33-o 
41.2 

17 

18 
19 

9.77616 
.77635 
.77654 
.77674 
.77693 

19 
19 

19 

10.17112 
.17166 
.17209 
.17257 
.17305 

46 
48 

48 

48 

9.78765 

.78785 
.78804 
.78823 
.78842 

19 
19 
19 
19 
19 

10.20028 
.20077 
.20126 
.20175 
.20224 

49 
49 
49 
49 
49 

15 

16 

17 
18 

19 

6 

I 

9 

10 

49 

4.9 

5-7 
6  I 

11 

48 

4-8 
5-6 
6.4 

20 

21 

22 
23 
24 

9.77712 
•77732 
.77751 
.77776 
.77790 

I9 
19 
19 
i§ 
19 

10.17353 
.17401 

•17449 
.17498 

.17546 

.  0 

9.78861 
.78886 
.78899 
•78918 
.78937 

19 
19 
19 

10.20273 
.20323 
.20372 
.2042! 
.  20476 

49 
49 
49 
49 
49 

20 

21 

22 
23 
24 

20 
30 
40 
50 

16.3 
24.5 

40-8 

16.1 
24.2 

3«-3 
40.4 

25 
26 

27 
28 

29 

9.77809 
.77828 
.77847 
.77867 
.77886 

i§ 
19 
i§ 
i§ 

10.17594 
.17642 
.17696 

•17739 
.1778? 

48 
48 
48 
48 
48 

,Q 

9.78956 
.78975 
.78994 
.79013 

.79032 

19 
19 
19 
19 

10.20520 
.  20569 
.20618 
.20668 
.2071? 

49 
49 
49 
49 
49 

25 
26 
27 
28 
29 

6 
I 

48 

4.8 
5-6 
6.4 

4-7 

3:j 

30 

3' 

32 
33 
34 

9.77905 
.77925 
•77944 
.77963 
.77982 

19 
19 
19 

10.17835 
.17884 
.17932 
.17986 
.18029 

46 

48 
48 
48 

48 

9.79051 
.79069 
.79088 
.79107 

.79126 

J9 
i§ 
19 
19 
19 

10.20767 
.20816 
.20865 
.20915 
.20964 

49 
49 
49 
49 
49 

A  A 

30 

32 

33 
34 

9 
10 

20 
30 
40 
50 

7.2 
8.0 
16  o 
24  o 
32  o 
40.0 

7-9 
15-8 
23.7 

3'-6 
39-6 

P 

37 
38 

!  39 

9.78002 
.78021 
.  78046 
-78059 
.78078 

!9 

19 

10.  1807? 

.18126 

.'18222 
.18271 

! 

9-79U5 
.79164 

.79183 
.79202 
.79220 

19 
19 
18 

10.21014 
.21063 
.21113 
.21162 

.21212 

49 
49 
49 
49 
5° 

36 

38 
39 

6 

1.9 

19 

1.9 

40 

42 
43 
44 

9.78098 
.78117 
.78136 
.78155 
.78174 

19 
'9 

*9 

19 
Tft 

10.  18319 

'.18465 
.18514 

48 

9.79239 

•79258 
.79277 

.79296 
.79315 

19 
19 
18 
19 
19 

10.21262 
.21311 
.21361 
.21410 
.21466 

49 
49 
49 
49 
50 

i  A 

40 

42 
43 

44 

7 
8 
9 
10 

20 
40 

2.3 

2.6 

2.9 

3-2 

6.5 
9-7 
13.0 

2.2 

6-3 
9-5 

12  6 

45 
46 
47 
48 
49 

9.78194 
.78213 
.78232 
.78251 
.78276 

J9 

i§ 
19 

10.18562 
.18611 
.18659 
.18708 
.18757 

48 
48 
48 
48 
49 

9-79333 
.79352 

•79371 
.79390 

.79409 

19 
19 

I0.2I5IO 

.  2  1  560 
.21609 
.21659 
.2I7C9 

49 
50 
49 
50 
49 

45 
46 
47 
48 

49 

50 

J5-8 

50 

52 
53 
54 

9.78289 

•78309 
•78328 

.78347 

19 

19 
19 

10.18805 
.18854 
.18903 
.18951 

.19006 

48 
48 
49 
48 
49 

9.7942? 
•79446 
.79465 
.79484 
.79503 

19 
19 
18 

19 

10.21759 
.21808 
.21858 
.21908 
.21958 

5° 
49 
50 

5° 
49 

50 

52 
53 
54 

: 

6 
7 
8 
9 

10 
20 

V.I 

2.4 

2.8 

1:1 

11 

57 
58 
59 

9.78385 
.78404 

•78423 
.78442 
.  78462 

19 
19 
19 

IQ 

10.  19049 

.19098 
.19145 
.19195 
.  19244 

48 
49 
48 
49 
49 

AQ 

9.79521 
.79546 

•79559 
•79578 
•79596 

18 

10.22008 
.22058 
.22108 
.22158 
.  22208 

50 
50 
50 
50 

P 

57 
58 
59 

40    i 

50    i 

\'l 

60 

9.78481 

10.19293 

4o 

o.7Q6i5 

5 

10.22258 

j° 

tiO 

LOB.  Vers. 

-D 

Log.  Exser. 

I) 

LOJT.  Vers.  1 

n 

Log.  Exser. 

n 

P.  P. 

427 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL    SECANTS. 

68°  69° 


/ 

Log.  Vers. 

D 

Log.  Exseo. 

JD 

Log.  Vers 

D 

Log.  Exsec. 

D 

/ 

P. 

P. 

0 

2 

3 

4 

9.79615 
.79634 
.79653 
.79671 
.79690 

18 
19 
18 
18 

IO.22258 
.  22308 
.22358 
.22408 
.22458 

50 
50 
50 
50 

9.80723 
.80747 
.80765 
.80783 
.  80802 

18 
18 
18 

18 

10.25295 
.25347 

•25398 
.25449 
.25501 

It 

51 

5i 
_~ 

0 

I 

2 

3 
4 

6 

7 

5 

5 
6 

3 

^ 

•2 

55 

5'2 

5.8 

1 

7 
8 

9 

9.79709 
•7972? 

•7974S 
.79765 
.79783 

19 
18 
19 

IO.22508 
.22558 
.  22608 
.22658 
.2270§ 

5° 
50 
55 
50 
5o 

9.80826 
.80839 
.80857 
.80875 
.80894 

T  0 

*8 
J8 
18 

i§ 
18 

T  Q 

10.25552 
.2560^; 

•25655 
.25707 

•25758 

51 
5? 
5i 

S 

I 

8 
9 

8 
9 

JO 

ao 

30 
40 
50 

7 
7 
8 

J7 

26 

'35 
44 

O 

6 

5 

i 

7.0 

7-9 
8.7 

2«a 

26.2 

35-° 
43-7 

10 

ii 

12 

13 
14 

9.79802 
.79821 
.79839 
•79858 
.79877 

19 
i§ 
ij 
19 

10.22759 
.22809 
.22859 
.22909 
.22960 

5s 
50 
50 

5° 
5o 

9.80912 
.  80936 
.80949 
.80967 
.80985 

16 

18 
18 
18 

18 

T  C 

10.25810 
.25861 

•25913 
.25964 
.26016 

I 
5l 

5i 

5? 

_  c 

10 

ii 

12 

13 
14- 

6 

7 

5 

5 
6 

2 

2 

6 

ss 

5-i 
6.0 

11 

17 

18 
19 

9.79895 
.799H 
.79933 
.79951 
.79970 

T  0 

18 
18 
19 

IO.230IO 
.  23066 
.23116 
.23161 
.23211 

5° 
53 
50 

5° 
50 

9.81003 
.81022 
.81046 

.81058 
.81077 

lo 
18 
18 
18 

18 

T  8 

10.2606? 
.26119 
.26171 
.26222 

.26274 

51 

52 
5? 
5? 
52 

15 
16 

17 

18 
19 

9 

10 

20 
3° 
4« 
So 

s7 
17 
26 

34 
43 

S 

8 

i 

9 

0 

\ 

3 

•8 
7-7 
8.6 
17.1 
25-7 
34-3 
42.9 

20 

21 
22 
23 

24 

9.79988 
.  80005? 
.80026 
.80044: 
.80063 

T  0 

J8 
19 
i§ 
18 
18 

TO 

10.23262 
.23312 
.23362 
.23413 
.23463 

55 
56 
50 
So 
50 

rr» 

9.81095 
.81113 
.81131 
.81150 
.81168 

lo 

18 
18 

18 
18 

10.26326 
.26378 
.  26429 
.26481 

•26533 

5* 

8 

52 
52 

20 

21 

22 

23 

24 

6 

7 
8 

5 

5 

I 

i 

50 

5-o 
5-? 
6.7 

3 

27 
28 
29 

9.80081 
.80106 
.80119 
.8013? 
.80156 

1S 
19 

i§ 
18 
18 

10.23514 
.23564 

.23615 
.23666 

.23716 

^ 

$° 
50 

50 

5° 

rA 

9.81185 
.81204 
.81223 
.81241 
.81259 

*8 

18 

i§ 
18 

i§ 

T  8 

10.26585 
.26637 
.26689 
.26741 
.26793 

I? 

52 
52 
52 

25 
26 

27 
28 

29 

9 
10 

20 
30 
40 

5° 

17- 
25- 
34- 
42. 

6 

5 

0 

5 

o 

5 

7.6 

8-1 
16.  g 
25.2 
33-6 
42.1 

30 

3i 
32 
33 
34 

9.80174 
.80193 
.80211 
.80230 
.80243 

T  0 

*8 

'§ 

!§ 
18 

10.23767 
.2381? 
.23868 
.23919 
.23969 

5° 
50 
5i 

5° 
56 

9.8127? 
.81295 
.81314 
.81332 
.81356 

lo 

18 

18 
18 

18 

.0 

10.26845 
.26897 
.  26949 
.27001 
.27053 

52 
52 
52 
52 
52 

30 

3i 

32 
33 
34 

6 

7 
8 

50 

5-o 

H 

6-6 

35 
36 

P 

39 

9.8026? 
.80286 
.  8030^ 
.80323 
.80341 

I9 
'1 

ii 

¥  8 

IO.24020 
.24071 
.24122 
.24172 
.24223 

51 

5o 

I* 

51 

9.81368 
.81386 
.81405 

.81423 
.81441 

lo 
18 

18 
18 
18 

TO 

10.27105 

.2715? 
.27209 
.27261 
•27314 

52 
52 
52 
52 
52 

3 

11 

39 

9 
10 

20 

3° 

40 
50 

T 
1 

3 

4 

7-5 
8-3 
6.6 
5-0 
3-1 
1-6 

40 

41 
42 

43 
44 

9  .  80360 
.80373 
.80397 
.80415 
.  80434 

18 
J8 

ii 
18 

10.24274 

.24325 
.24376 
.24427 
.24478 

51 
5o 
5i 
5i 
5i 

9.81459 

.81477 
.81495 

.81513 
.81532 

J8 
18 
18 
18 
18 

T  C 

10.27366 
.27418 
•27470 
.27523 
•27575 

52 
52 
52 
52 
52 

40 

4i 
42 
43 

44 

6 
7 
8 

I( 

I. 

2. 
2. 

) 
.) 
1 

18 

*i 

2.1 
2.4 

2  8 

45 
46 

47 
48 

49 

9.80452 
.  80476 
.  80489 
.8050? 
.80526 

lo 

ii 
i§ 

T  R 

10.24529 
.24580 
.  2463  1 
.24682 
.24733 

51 
5i 
5i 
5i 
5i 

9.81550 
.81568 
.81586 
.81604 
.81622 

lo 
18 
18 

18 
18 

T  8 

10.2262? 
.  27680 
.27732 
.27785 
•2783? 

^ 
52 
52 
52 
52 
ro 

45 
46 

47 
48 

49 

10 

20 
30 
40 
5° 

3- 
6. 
9- 

12. 

IS- 

3 

i 

3 
j 

3-1 

6.1 
9.2 

12-3 

15-4 

50 

5i 

52 
53 

54 

9.80544 
.80563 
.8058? 
.80600 
.80618 

IS 

'i 

i§ 
18 

T  8 

10.24784 

.24835 
.  24886 

•24937 
.24983 

51 
51 
5i 

11 

9.81646 
.81658 
.81675 
.81695 
.81713 

lo 

18 
18 

18 
18 

T  R 

10.27890 
.27942 
.27995 
.  2804? 
.28100 

52 
55 

9 

52 

50 

5i 
52 

53 

54 

6 
7 

8 
9 

] 
•j 

: 

8 

.8 
.1 
•4 
•  7 

11 
% 

59 

9.80635 
.80655 
.80673 
.  80692 
.80710 

§ 

ii 

18 
18 

10.25039 
.25096 
.25142 

.25193 

.25244 

51 

5i 
5i 
51 
5? 

r  i 

9.81731 
.81749 
.81767 
.81785 
.81803 

18 
18 
18 
18 

JO 

10.28152 
.28205 
.28258 
.28316 
.28363 

52 
53 
52 
52 

$ 

55 
56 

P 

59 

10 

20 
3° 

4° 
5» 

i 
c 
\i 
if 

.0 

.0 

.0 
.0 
.0 

60 

9.80728 

10.25295 

31 

9.81821 

10.28416 

GO 

' 

Log.  ^  ers. 

D 

iOff.  Ex  sec. 

/> 

Loe.  Vers. 

j> 

jOtf.  ExRPC.l 

D 

' 

P. 

r. 

428 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL   SECANTS. 

7O°  71° 


Log.  Vers. 

Loar.  Exsec. 

It 

Loir.  Vers. 

LOST.  Exsec. 

D 

p. 

0 

9.81821 

TQ 

10.28416 

9.82894 

T7 

10.31629 

^ 

0 

I 

.81839 

ift 

.28469 

„ 

53 

5s 

.82911 

17 

.31684 

54 

r  ? 

i 

3 

.81857 

18 
iR 

.28521 
.28574 

2 
53 

.82929 
.82947 

I/ 

18 

•31738 

54 

54 

2 

5 

§ 

56 

4 

•81893 

i  P. 

.28627 

53 

.82964 

i  P 

•3184? 

54 

r  7 

4 

6 
7 

5 
6 

'6 

5-6 
6-5 

6 

9.81911 
.81929 

id 
18 

i  X 

10.28680 
.28733 

53 

9.82982 
.83000 

lo 

If 

10.31902 
.3^956 

54 
54 

i 

8 
9 

10 

9 

•5 
•  5 
.4 

7-4 

8 
9 

.81947 
.81965 
.81983 

18 
18 

T  Q 

.28786 
.28839 
.28892 

53 
53 
53 

.83017 
.83035 
•83053 

1f 

18 
if 

.32011 
.  32066 
.32126 

54 

55 
54 

7 
8 

9 

20 
30 
40 
50 

18 
28 
37 
47 

•i 

.2 

•6 
.  i 

iS.g 
28.0 
37-3 
46-6 

10 

ii 

9.82001 
.820I§ 

lo 

18 
iP, 

10.28945 
.  28998 

53 
53 

9.83076 
.83083 

i  n 

17 
18 

1  9 

10.32175 
.32230 

54 
55 

10 

ii 

12 

.8203? 
.82055 

18 
,c 

.29051 
.29104 

53 

53 

.83106 
.83123 

•7 
if 

.32284 
.32339 

54 
55 

CA 

12 
13 

6 

5 

5 

55 

5-5 

14 

.82073 

.29157 

53 

.83141 

if 

To 

•32394 

54 

H 

7 

6 

•  5 

6.4 

15 

16 

17 

9.82091 
.82109 
.82127 

17 
18 

10.29210 
.29263 

•293'6 

53 

53 

53 

9-83I59 
.83176 
.83194 

if 

•  I? 

10.32449 
.32504 
•32558 

55 
55 
54 

15 

16 

17 

9 
10 

20 

3° 

9 
18 
27 

•3 

.2 

•5 

9-i 
27.5 

18 

.82145 

TQ 

.29370 

3 

.83211 

lf 
IS 

.32613 

55 

18 

40 

.0 

36-6 

19 

.82163 

,0 

•29423 

53 

.83229 

.  32663 

55 

19 

5« 

40 

.2 

45-8 

20 

9.82,81 

TQ 

10.29475 

53 

9.83247 

•  4 

10.32723 

55 

20 

21 

.82199 

i  9 

•29529 

53 

.83264 

1  / 

•32778 

55 

21 

22 
23 

.82217 
•82235 

18 

T7 

.29583 
.29636 

53 
53 

.83282 
.83299 

17 

17 

,0 

•32833 
•32888 

55 
55 

rg 

22 
23 

6 

5 

S 

54 

5-4 

24 

.82252 

17 
T0 

.  29689 

? 

.83317 

•32944 

55 

24 

7 

8 

•3 

6-3 

25 

9.8^276 

10.29743 

53 

50 

9-83335 

lf 

10.32999 

55 

25 

9 

8 

.2 

8.1 

26 
27 

.82288 
.82305 

18 

,0 

•29796 
.29850 

3 
53 

.83352 
•83370 

If 

.33054 

55 
55 

26 
27 

10 
20 

9 
18 
27 

.  I 

.1 

.2 

Q.O 

18.0 
27.0 

28 
29 

.82324 
•82342 

If 

T  8 

.29903 
•29957 

53 
53 

•83387 
•  83405 

If 

•33164 
.33220 

55 
55 

28 
29 

40 

5° 

36 
45 

•3 
•4 

36.0 
45  -° 

30 

9.82360 
.823/8 

lo 

18 

TQ 

10.30016 
.30064 

53 

9.83422 
.83440 

If 

10.33275 
•33336 

55 
55 

30 
31 

32 

.82396 

17 

.30117 

53 

.83458 

l$ 

•33385 

55 

32 

5 

3 

53 

33 
34 

•82413 
.8243? 

17 

18 

T  0 

.30171 
.30225 

\\ 

.83475 
.83493 

If 

•33441 

•33496 

55 
55 

cp 

33 
34 

6 
7 
8 

1 

7 

•I 

7:| 

35 
36 

9.82449 
.82467 

17 

iX 

10.30273 
.30332 

53 
54 

9.83516 
.83528 

If 

10.33552 

55 
55 

rg 

35 
36 

y 

10 

20 

8 
8 
17 

0 

•9 
I 

J:| 

% 

39 

.82485 
.82503 
.82526 

18 
17 

i  S 

.30386 
•30440 
.30493 

53 
54 
53 

.83545 
.83563 
.83586 

If 
If 

I  ^7 

.33663 
•33718 
•33774 

55 

11 

r  p 

37 
38 
39 

3° 
40 
5° 

26 

35 
44 

•7 

35-| 
44-1 

40 

9.82538 

15 

,  Q 

10.30547 

54 

9.83590' 

17 

10.33829 

55 

40 

4i 
42 

3 

•82556 
.82574 
.82592 
.82609 

If 

18 

i  Q 

.30601 
•30655 
.  30709 
.  30763 

53 
54 
54 
54 

.83615 
.83633 
.83656 
.  83668 

1  7 
if 
if 

If 

.33885 
•33941 
•33996 
.34052 

55 
55 
56 

_  p 

42 
43 
44 

6 
7 
8 
9 

i 

S 

7.0 

47 

9.82627 

.82645 
.82663 

lo 
18 
if 

T(Q 

10.30817 
.30871 
•30925 

54 
54 
54 

9.83685 

.83703 
.83720 

I? 
11 

10.34108 
.34164 
•34220 

55 
56 
56 

r? 

45 
46 
47 

10 

20 
3° 
40 

i 

2( 

3. 

7'5 
5.2 

>-o 

48 
49 

.82681 
.82693 

17 

T  Q 

.30979 
•31033 

54 

54 

.83737 
.83755 

If 

i  fj 

.34275 
•34331 

55 
56 

r  A 

48 
49 

5° 

4. 

J-7 

50 

9.82715 

lo 
17 

10.31087 

54 

9.83772 

17 

17 

10.34387 

56 

50 

52 
53 
54 

•82734 
.82752 
.82769 
.82787 

1  / 

18 

17 
18 

jC 

.31141 

•3^95 

.31249 
•31303 

54 
54 
54 
54 

.83790 
.83807 
.83825 
.83842 

J  / 

!? 
if 

•  34443 
•34499 
•34555 
.34611 

56 
56 
56 
cf. 

52 
53 

54 

6 
7 
8 
9 

18 

1.8 

2.1 

2.4 

2.7 

2.6 

17 

7 

0 
2 

5 

55 
56 

9.82805 
.82823 
.82846 
.82853 

18 

if 
18 

10.31358 
.31412 

•31465 
.31521 

54 
54 
54 
54 

9.83859 
.83877 
.83894 
.83912 

17 

Tfl 

10.34667 

•34723 
•  3478o 
•34836 

5° 
56 
56 
56 

P 

57 
58 

10 

20 

30 
40 
5° 

i 
i 

).O 

z.o 
5.0 

ii 

ii.  g 
I4.6 

1 
8  5 

14.1 

59 

.82876 

18 

•31575 

•83929 

1  / 

I  7 

•  34892 

r2 

59 

|  00 

9.82894 

10.  31629 

jl 

9-83946 

10.34948, 

DO 

GO 

Loir.  Vers. 

.oar.  Exsec. 

Locr.  Vers. 

J> 

' 

p. 

P. 

TABLE    VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL    SECANTS. 

12°  73° 


/ 

Log.  Vers. 

n 

Log.  Exsec. 

Log.  Vers. 

JD 

Log.  Exsec. 

; 

P.  P. 

0 

9.83946 

T« 

10-34948 

„ 

9  .  84986 

10.38387 

^ 

0 

I 

.83964 

17 
*a 

.35005 

c5 

.8499? 

I7 

.38445 

58 

i 

2 

.83981 

17 

.35061 

5° 

C2 

.85014 

.38504 

58 

2 

3 

.83999 

•351^ 

If 

.85031 

I7 

.38562 

8 

3 

61 

65 

4 

.84016 

•7 

T  fj 

.35174 

56 

.85049 

I/ 

.38621 

58 

4 

6 

6. 

6.0 

'  6 
7 

9-84033 
.  8405  1 
.84063 

If 
If 
If 

10.35230 
.35286 

•35343 

5§ 
56 

56 

9.85066 
.85083 
.85100 

17 
17 

17 

10.38679 
.38738 
.38796 

58 
58 

6 

7 

8 
9 
10 
20 

8. 
9- 

10. 

20.3 

8.5 
9.1 

IO.  I 
20.1 

8 
9 

.84085 
.84103 

17 
I? 

,G 

•35399 
•35456 

56 

57 

.85117 
•85134 

I7 
17 

.38855 
.38914 

59 
58 

8 
9 

3° 

40 
So 

30.5 

soli 

30.2 
40.3 
50-4 

10 

ii 

12 
13 

9.84126 

.8413? 
.84155 
.84172 
.84189 

17 
17 
If 
If 

17 

.35626 
.35683 
•35739 

56 
56 

57 
56 

9.85151 
.85168 
.85185 
.85202 
.85219 

I7 

17 
17 
17 
17 

10.38973 

•39031 
•39090 

.39H9 
.  39208 

59 
58 
59 
59 
58 

10 

ii 

12 

13 
U 

6 
7 

60 

6.0 
7.0 

59 

5-9 
6.8 

15 

9.84207 

If) 

10-35796 

57 

9.85236 

17 

10.39267 

59 

IS 

8 
9 

8.0 
9.0 

7-9 
8.9 

16 

.84224 

17 

.35853 

56 

•85253 

I7 

.39326 

59 

16 

10 

IO.O 

9.9 

17 
18 

19 

.84241 
.84259 
.  84276 

I? 
17 

.359'o 

.35967 
.36023 

57 
57 
56 

.85270 
.85287 
•85304 

17 
17 

•39383 

•39444 
•39503 

59 
59 
59 

i7 
18 

19 

20 

3° 
40 

5° 

20.0 
30.0 
40.0 
50.0 

19.  § 
29.7 
39-6 
49.6 

20 

9.84293 

17 

To  .  36086 

57 

9.85321 

17 

10.39562 

59 

20 

21 

.84316 

T7 

.36137 

57 

.85338 

J7 

.39621 

59 

21 

22 
23 
24 

.84328 

.84345 
.84362 

17 
17 
17 

.36194 
.3625? 
•36303 

57 
57 
57 

_fi 

.85355 
•85372 
•85389 

17 
17 
17 

.39681 
•  39740 
•  39799 

59 
59 
59 

22 

23 
24 

6 

7 
8 

59 

5-9 
6.9 
7.0 

58 

5-8 
6.8 
7-8 

25 
26 

27 
28 

29 

9.84380 

.84397 
.84414 

.84431 
.84449 

17 
If 

17 

10.36366 

•36423 
.  36480 

.3653? 
•36594 

57 
57 

5 

57 

["9 

9.85405 
.85422 
•85439 
.85456 
.85473 

17 
17 
17 
17 

10.39859 
.39918 

•3997? 
.40037 
.40095 

fr\ 

59 
59 
59 
59 
59 

25 
25 

27 
28 
29 

9 

10 

20 
30 
40 
50 

8. 

,9'§ 
29-5 
39-3 
49.1 

8.8 

9-7 
39-5 
29.2 
39-o 
48.7 

30 

9  .  84466 

17 
T  7 

10.36652 

57 

9.85496 

17 

10.40156 

59 

f\Ct 

30 

31 

.84483 

1  / 

•  36709 

57 

rf} 

•8550? 

J7 
T? 

.40216 

31 

32 
33 
34 

.84506 
.8451? 
.84535 

17 

•36765 
.36824 
.36881 

57 

3 

.85524 
.85541 
.85558 

T6 
17 
17 

•40275 
•40335 
.40395 

59 
59 
60 

32 
33 
34 

6 

I 

5» 

5-8 

7-7 

?\ 

I'i 

3 

37 
38 

9.84552 
.84569 
.84585 
•  84603 

I? 
17 
17 

10.36938 
.36996 

.37054 
.37111 

57 

I 

9-85575 
.85592 
.85603 
.85625 

17 
17 
16 
17 

10.40454 
.405H 
.40574 
.40634 

59 
60 

59 
60 

11 

37 
38 

9 

10 

20 

3° 
40 
.So 

8.7 

'9-3 
29.0 

8.6 
9.6 
19.1 
28.7 
38.3 
47-9 

39 

.84626 

!7 

T  <7 

.37169 

Pfj 

.85642 

17 

.40694 

/• 

39 

40 

9.84638 

17 

10.37225 

57 

9.85659 

17 

10.40754 

DO 
/;„ 

40 

42 

43 
44 

.84655 
.84672 
.  84689 
•84705 

17 
17 
17 

.37284 
.37342 
•37399 
•3745? 

58 
57 
58 

r  o 

.85676 
.85693 
.85710 
•85725 

16 
17 
17 
16 

.40814 
.40874 
.40934 
.40994 

60 
60 
60 

42 
43 
44 

6 

7 
8 
9 

57 

5-7 
6.  g 
7.6 
8.5 

56 

5-6 
6.6 

R':5; 

45 
46 

9.84724 
.84741 

17 

•37573 

s 

CQ 

9.85743 
.85766 

17 

17 

10.41054 
.41114 

60 

45 
46 

10 

20 

3° 

9-5 

IQ.O 
28.5 

9-4 

18.5 

28.2 

47 
48 

.84758 
.84775 

1   / 

If 

.37631 
.37689 

5° 
58 

CQ 

.85777 
.85794 

17 

.41174 
.41235 

66 

47 
48 

40 
50 

38.0 

47-5 

37-6 

49 

.84792 

17 

.37747 

5° 

ro 

.85811 

17 

.41295 

49 

50 

9.84809 

17 

10.37805 

58 

CQ 

9.8582? 

Jo 

10.41355 

DO 

50 

52 
53 

.84825 
.  84844 
.84861 

17 
if 
17 

T  7 

.37863 
.37921 

•37979 

55 
5? 

55 

.85844 
.85861 
.85878 

17 
17 
16 

.41416 
•4H76 
•41537 

60 

66 

fart 

52 
53 

6 
I 

If    I? 

1.7   1.7 

2.O    2.C 

1-6 
i.g 

54 

.84878 

1  / 

.3803? 

8 

.85895 

I7 

.4159? 

54 

9 

2.§    2.2 

2.6   2.5 

2.5 

55 
56 

57 

9.84895 
.84912 
.84929 

17 
'7 

10.  38095 

•38153 
.38212 

58 
58 

9.85911 
.85923 
.85945 

17 
17 

10.41658 
.41719 
•41779 

bo 
6.1 

66 

6n 

55 
56 
S7 

10 
20 
30 

40 

i 

2.0    2.£ 

5-8   5-6 
8.7   8.5 

2.7 

5-5 

8.2 
II.  0 

58 

.84945 

1  / 

.38276 

58 

.85962 

Z6 

.41840 

fii 

58 

5° 

4.6  14.1 

I3'7 

59 

.84963 

1  / 

17 

•38323 

C8 

.85979 

i  / 

1  A 

.41901 

61 

59 

60 

9.84986 

10.38387 

JO 

9.85995 

Jb 

10.41962 

(JO 

; 

Log.  >  ers. 

7> 

Loar.  Kxsec. 

__  -"._.! 

Lotr.  V«TH. 

J> 

josr.  KXSPC. 

7> 

' 

P.  P. 

430 


TABLE   VIII. —LOGARITHMIC    VERSED   SINES   AND    EXTERNAL   SECANTS. 

74°  75° 


j  / 

Los.  Vers. 

Log.  Exsec. 

D 

Loer.  Vers. 

Lotf.  E.\Hec. 

p 

P. 

0 

9.85995 

10.41962 

An 

9.86992 

10.45693 

A3 

0 

I 

.86012 

1  / 

.42022 

.87009 

'6 

.45756 

°3 
A3 

i 

2 

3 
4 

.86029 
.86046 
.86062 

17 

'6 

.4208^ 
.4214^ 
.42205 

6l 
6l 
Ai 

.87025 
.87042 
.87058 

'6 
16 

.45820 
.45884 
•4594? 

03 
64 

63 
f.. 

2 

3 

4 

6 
7 

67 

6.7 

7-8 

68 

6.$ 
7-7 

66 

6.6 
7-7 

6 

9.86079 
.86096 

17 
16 

10.4226^ 
.4232? 

61 

9.87074 
.87091 

16 

10.46011 

.46075 

°4 
64 
A/i 

6 

8 
9 

10 

8.9 

10.  0 

ii.  i 

8.8 

10.  0 

II.  I 

8.8 
9-9 

II  0 

8 
9 

.86113 
.86l2§ 

.86146 

17 
16 
17 

•42388 
.42450 
.42511 

61 
61 

AT 

.87107 
.87124 
.87146 

'6 

.46139 
.  46203 
.  4626? 

04 
64 

64 
s  . 

8 
9 

20 

30 
40 

5° 

22.3 
33-5 

22.1 

33-2 
44-3 
55-4 

22.  0 

33-o 
44.0 
55>0 

10 

9.86163 

T6 

10.42572 

OI 
AT 

9.87157 

10.46331 

64 
A/I 

10 

ii 

.86179 

]6 

•42633 

AT 

.87173 

.46395 

04 
A/f 

1  1 

12 
14 

.86195 
.86213 
.86230 

16 
17 

•42695 
.42756 
.42817 

61 
61 

AT 

.87189 
.87206 
.87222 

16 

16 
.  p 

.  46460 
.46524 
.46588 

04 
64 
64 
A/« 

12 
13 
14 

6 

7 

6-I 

ol 

^5 

64 

6.4 
7-5 
8  6 

15 

9.86245 

i2 

10.42879 

OI 

AT 

9.87239 

'6 

iA 

10.46652 

O4 

15 

9 

Ii 

9-7 

9-7 

16 

17 
18 

i  I9 

.86263 
.86280 
.86295 
•86313 

J6 
17 
'6 
16 

.42940 
.43002 
.43063 
•43125 

61 
6T 
62 

AT 

.87255 
.87271 
.87288 
.87304 

16 
16 
16 

.46717 
.46781 
.  46846 
.  469  i  6 

64 
64 
64 
64 
A? 

16 

17 
18 

19 

IO 
20 
30 
40 
50 

10.9 

21.  jj 
32-7 

43-6 
54-6 

10.  | 
21.6 

32.5 

43-| 

10.7 

2I-s 

32.2 
43  o 
53-7 

20 

21 
22 
23 
24 

9.86330 

.86346 
.86363 
.86380 
.86396 

ii 
17 

16 

10.43187 

•43249 
•43310 
•43372 
•43434 

62 

6T 
62 
61 

9.87320 
.87337 
•87353 
.87370 
.87386 

16 

16 
16 

10.46975 
.  47040 
.47104 
.47169 
.47234 

°4 
65 
64 
65 

u 

A  r 

•20 

21 
22 
23 
24 

6 
7 

8 

6.4 
7-4 
8-5 

63 

6-3 

63 

26 

9.86413 
.  86430 

17 

10.43496 
•43558 

62 

9.87402 
.87419 

tf 

tJfi 

10.47299 
.47364 

65 

65 

Ar 

25 
26 

9 

IO 
20 

9.6 
21.  § 

9  5 
10.6 

21.  I 

9-4 

21.0 

!  27 
28 
29 

.86446 
.86463 

.86479 

J6 
16 
'6 

.43620 
.43682 
•43744 

62 
62 

A3 

.87435 
.87451 
.87468 

16 
16 

T  A 

.47429 
•47494 
•47559 

°5 

65 

65 
Ar 

27 
28 
29 

30 
40 

5° 

32  o 
42-6 
53-3 

42.§ 
52.9 

3J.5 
42.0 
52.5 

30 

9.86495 

17 

10.43805 

9.87484 

IO 

Tp 

10.47624 

65 

30 

32 
33 
34 

.86513 
.86529 
.86546 
.86562 

5i 

•43863 
•43931 
•43993 
•44055 

62 
62 
62 
A3 

.87506 
•87516 
•87533 
.87549 

*o 

16 

16 
16 

T  A 

•47689 

•47754 
.47820 
.47885 

65 

65 

65 

6f 

31 
32 
33 
34 

6 
7 
8 

62 

6.2 

7.3 

8.3 

62 

6.2 

1:1 

6i 

t: 

35 
36 
37 

9.86579 
.86596 
.86612 

17 

10.44118 
.44180 
.44242 

O2 
62 
62 
A3 

!  87582 
.87598 

IO 

'6 
16 

10.47956 
.48016 
.48081 

5 
65 

65 

AP 

35 
36 

37 

10 

20 
30 
4° 

10.4 

20.  § 
31-2 
41-6 

20.6 

31.0 

10.2 
20.5 
30.7 
41.0 

38 

.86629 

9 

.44305 

fo 

.87614 

16 

.48147 

3« 

50 

52.1 

51-6 

5'  -2 

39 

.86645 

_  p 

.44368 

°3 

A3 

.87631 

•48213 

39 

40 

9.86662 

10.4443° 

A3 

9.87647 

IO 

10.48273 

5 

AP 

40 

42 

.86678 
.86695 

17 

•44493 
.44556 

63 

6  a 

.87653 
.87679 

16 

.48344 
.48410 

66 
66 

42 

6 

4 

6 

Si 

.1 

66 

6.6 

43 

.86712 

9. 

.44613 

A-» 

.87696 

?A 

.48476 

AA 

43 

I 

I 

.1 

7.0 

44 

.86723 

*6 

.44681 

°3 
63 

.87712 

.48542 

Ar 

44 

8 
9 

Q 

.1 

8.6 
9.1 

45 
46 

9.86745 
.86761 

16 

10.44744 
.44807 

2 
63 

9.87723 
.87744 

16 

10.4860? 
.48674 

°5 

66 

66 

45 
46 

10 

20 
3° 

1C 

20 

30 

.1 

•3 
•  5 

10.  1 

20.1 
30.2 

47 

.86778 

.44870 

Ao 

.87761 

i  A 

.48740 

AA 

47 

40 

40 

•0 

40-3 

48 

.86794 

r§ 

•44933 

63 

A-7 

.87777 

tA 

.48806 

AA 

48 

50 

5C 

•8 

50.4 

49 

.86811 

.44996 

°3 

.87793 

.48872 

49 

50 

9.8682? 

I  A 

10.45059 

9.87809 

TA 

10.48933 

66 

50 

52 
53 
54 

.86844 
.86866 
.86877 
.86893 

IO 

16 
16 
16 

.45122 
.45185 
•45248 
•45312 

63 
63 
63 

A3 

.87825 
.87842 
.87858 
.87874 

16 
16 

*6 

T  A 

.49004 
.49071 

.49137 

.49204 

66 
66 
66 

52 
53 
54 

6 

7 
8 

Q 

2.O 
2.2 

2-5 

16 

•6 
9 

.2 

•5 

16 

1.6 

2.1 

2-4 

1 

9.86910 
.86925 
.86943 
.86959 

!6 
16- 

,p 

10.45375 
.45439 
.45502 

•45565 

63 
63 
63 

6? 

9.87896 
.87905 
.87923 
.87939 

IO 

16 

16 
16 

TA 

10.49270 
•49337 
.49403 
.49476 

67 

1 

IO 
20 

3° 

40 

50 

"•3 
14.1 

•  7 

1:3 

II.O 

| 

59 

.86976 

'6 

IA 

.45629 

°3 

6A 

.87955 

IA 

•49537 

6? 

59 

60 

9  .  86992 

^o 

10.45693 

9.8797! 

fo 

10.49604 

60 

LOST.  Vers. 

jOi?.  Exsec. 

D 

Lop.  Vers. 

>off.  Exser.l 

' 

P. 

P. 

431 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL    SECANTS. 

76°  77° 


' 

Log.  Vers. 

[MS;.  Kxsec. 

D 

Loar.  Vers. 

J» 

Lo?.  Exsec. 

D 

/ 

P. 

p. 

0 

2 

3 

•  4 

9.8797! 

.87987 
.88003 
.  88020 
.88036 

16 
16 

T  A 

i  o  .  49604 

.49676 

•4973? 
.49804 
.4987? 

66 

67 
67 

A<7 

9-88933 
.88949 
.88964 
.88986 
•88995 

16 

16 
16 

T  A 

10.53724 

•  53794 
.53865 
•  53936 
•  54007 

71 
70 
71 

0 

i 

2 

3 

4 

*7C 

T3 

6 

8 
9 

9.88052 
.88068 
.  88084 
.88100 
.88115 

IO 

16 
16 

10.49939 
.  50006 

.50073 
.50146 
.  50208 

07 

67 
67 

6? 

AT 

9.89012 
.89028 
.  89044 
.  89060 
.89075 

IO 

16 
16 
i$ 

T  A 

10.54078 
.54149 
.  54220 
.54291 
.  54362 

71 

7i 
7i 

6 

8 

9 

6 
7 

8 
9 

IO 
20 

75 

51 

10.  0 
II  .2 

I2.5 
25.0 

37.5 

7-4 
8.g 
9-8 
ii  .  i 

12.  i 
24-6 

37.0 

73 

9-7 
lo.o 

12.  I 

10 

ii 

12 

13 
14 

9.88133 
.88149 
.88165 
.88l8l 
.8819? 

16 

16 
16 
16 

T  A 

10.50275 
•  50342 
.50410 

•5047? 
.50545 

07 

6? 

6? 

68 

9.8909! 
.8910? 
.89123 
.89139 
.89155 

IO 

16 

16 
16 

_  a 

10-54433 
.54505 
•54576 
.  5464? 
.54719 

7* 

7? 
7i 
72 

T? 

10 

ii 

12 
13 

14 

40 

5° 

50.0 
62.5 

f'3 
61-6 

60.  8 

17 
i8 

19 

9.88213 
.88229 
.88245 
.88261 
.8827^ 

IO 

16 
16 
16 

10.50613 
.50681 

.  50748 
.50815 
.  50884 

68 

V 
68 
68 
AQ. 

9.89176 
.89185 
.  89202 
.89218 
.89234 

16 
16 

10.54791 
.54862 

•  54934 
.55006 
.55078 

71 

7? 

72 

71 

72 

II 

17 

18 
19 

6 
7 
8 

9 

10 

72 

K 

9.6 

10.8 

12.0 

71 

7-1 

ol 
10.  6~ 
ii.  5 

70 

7.0 

8.2 

9-4 
10.6 
11.7 

20 

21 
22 

23 
24 

9.88294 
.88310 
.88326 
.88342 
.88358 

'§ 
16 

16 
16 
16 

T  A 

10.50952 
.51026 
.51083 

.5H57 

.51225 

68 
68 

6g 
oo 

9.89249 
.89265 
.89281 
.89297 
.89312 

16 
16 

T  A 

10.55150 
.55222 
•55294 
•55366 
•55438 

72 
72 
72 

72 

20 

21 

22 

23 
24 

20 
3° 
40 

5° 

24.0 

36.0 
48.0 
60.0 

23-6 

35-5 
47-3 
59-1 

3s'i 
47.0 

58.7 

25 
26 

27 
28 

29 

9.88374 
.88390 
.  88406 
.88422 

.88438 

IO 

16 

5s 

16 

T  A 

10.51293 
.51361 
.5H30 

•51498 
.51567 

68 
68 
68 

68 

A/-» 

•  89344 
•  89360 
.89376 
•89391 

IO 

16 
16 
1$ 

10.55511 
•55583 
.55655 
•55728 
.55801 

72 
72 
72 
73 
72 

11 

27 
28 
29 

6 

7 

8 

69 

6.9 
8.6 

9.2 

68 

6.8 

7-9 
9  .0 

6.7 
8.6 

30 

32 
33 
34 

9.88454 
.88476 
.88485 
.88502 
.88518 

IO 

16 
16 
16 
16 

10.51636 
.51704 

•51773 
.51842 
.51911 

69 

68 
68 
69 
69 

An 

9.89407 
.89423 
•89438 
•89454 
.89470 

16 
IS 

T  A 

10.55873 

•55946 
.56019 
.  56092 
•56165 

•70 
72 

73 
72 
73 
73 

80 

31 
32 

33 
34 

9 

IO 
20 

3° 
40 

50 

10.3 
11  5 
23-0 
34-5 
46.0 
57-5 

10.2 

"••i 

22.  6 

34-o 
45-3 
56-6 

IO.O 

ii  .  i 

27.  3 

33-5 
44-6 
55-8 

35 
36 
37 
38 
39 

9.88534 
.88556 
.88565 
.88582 
.88598 

16 
16 
16 

T  A 

10.51980 

•52049 
.52118 
.52187 
.52256 

09 

69 
69 
69 

69 
AA 

9.89486 
.89501 
.8951? 
•89533 

•89548 

IO 

16 

Is 

i  A 

10.56238 
.56311 
.  56384 
•  5645? 
•5653' 

73 

73 
73 
73 
73 

35 
36 

37 
38 
39 

6 

t 
t 

>6 

6 

0.0 

40 

42 
43 
44 

9.88614 
.88636 

.88645 
.88662 
.88678 

IO 

16 
16 

\l 

T  A 

10.52325 

.52394 
.  52464 

.52533 
.  52603 

69 

6§ 
69 
69 

9.89564 
.89580 
.89596 
.89611 
.89627 

ID 

16 
IS 

T  A 

10.  56004 
.  56678 
.56751 
.56825 
.  56899 

73 
73 
73 
74 
73 

40 

42 
43 
44 

9 

IO 

20 

3° 
40 
5° 

i 
c 
i] 

25 

31 

4- 

5  = 

•7 
.8 
•9 
.0 
.0 

0 
.0 

.0 

0.6 

O.  I 
0.  I 
O.I 
O.2 

o.§ 

O.4 

4I 
46 

47 
48 
49 

9.88694 
.88710 
.88726 
.88742 
.88758 

IO 

16 
16 
16 

T  A 

10.52672 
.52742 
.52812 
.52881 
•52951 

70 
69 

70 

9.89643 
•  .89658 
.89674 
.89690 

•89705 

IO 

i§ 

iS 

16 

10.56973 

'57047 
.57126 

.57195 
•57269 

74 
74 
73 
74 
74 

45 
46 
47 
48 

49 

50 

52 
53 

54 

9.88774 
.88790 
.88805 
.88821 
.8883? 

IO 

16 

i5 
16 
16 

T  A 

10.53021 

•  53091 
.53161 

•53231 
•  533oi 

70 
70 
70 
70 

70 
_  ~ 

9.89721 
.89737 
.89752 
.  89768 

•89783 

16 

15 

10.57343 
•5741? 
.57491 
.57566 
.  57646 

74 
74 
74 
74 

50 

52 
53 

54 

6 

7 
8 
9 

IO 

20 

16 

1-6 
1.9 

2.2 

2-§ 

2.7 
5.5 

1  6 

1.6 

1-8 

2.  I 
2-4 

8:1 

5 

\ 

.6 

55 
56 
57 
58 
59 

9.88853 
.88869 
.88885 
88901 
.88917 

ID 
16 

16 
16 
16 

10.53372 

•  53442 
•53512 
•53583 
•53653 

70 
70 
7o 
76 
76 
76 
7 

9.89799 
.89815 
.  89836 
.  89846 
.89862 

10 

15 

T? 

10.57715 
•  57790 
•  57864 

•  57939 
.58014 

75 

74 
75 

75 

7  C 

55 

57 
58 
59 

3° 

40 

5° 

II  .0 

13.7 

10.0 

13.3 

10.3 
12.9 

60 

9.88933 

10.53724 

9-8987? 

1  3 

10.  58089 

/  :> 

00 

LOR.  Vers. 

Log.  Kxseo. 

7> 

Loir.  Vers. 

Lou.  Kxsec. 

f 

p 

p. 

432 


TABLE    VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL   SECANTS. 

78°  79° 


/ 

Lo*.  Vers. 

Mg.  Exsec. 

Log.  Vers. 

..ojr.  Exsec. 

P. 

P. 

0 

2 

3 
4 

9.89877 
.89893 
.89908 
.89924 

.89939 

_  a 

1  5 

T  f\ 

10.58089 
.58164 

.58239 
.58315 

.58390 

75 
75 
75 
75 

9.90805 
.90826 

.90835 
.90851 
.90865 

iS 

^5 

10.62745 
.62825 
.  62906 
.62985 
.63067- 

86 

86 
81 

op. 

0 

i 

2 

3 
4 

6 
7 

86 

8.6 
10.6 

I5, 

9  9 

I4, 

9.8 

1 

8 
9 

9.89955 
.89971 

.89986 
.90002 
.9001? 

ID 

Is 

10.58465 
.58541 

.58616 
.  58692 
.58768 

75 

75 
76 
75 

9.0088T 
.90897 
.90912 
.90927 
.90943 

i.S 

15 

10.63148 
.63229 
.63310 

.6339' 
.63472 

Si 
81 
81 
81 

RT 

6 

8 
9 

8 

9 

10 
20 
30 

4° 
5° 

11.4 

12.  q 

3:1 

43.0 

57-3 
71-6 

12.  jr 
14.1 
28.3 

56-6 
70.8 

II.  2 
12.6 

14.0 
28.0 
42.0 
56.0 
70.0 

10 

ii 

12 

13 
14 

9.90033 

•90048 
.90064 
.90080 
.90095 

16 

10.58844 
.  58920 

.58995 
.59072 
.59148 

75 
75 

76 

1f\ 

9.90958 
.90973 
.90983 
.91004 
.91019 

15 
15 

10.63553 
•63634 
.63716 

.6379? 
.63879 

81 
81 

81 
81 

10 

ii 

12 
13 

6 
7 

83 

8.3 

9-7 

82 

8.2 

9-1 

8l 

8.1 
9.4 

II 

17 
iS 

19 

9.90III 
.90125 
.90142 
.9015? 
.90173 

i5 

15 

Is 

10.59224 
.59306 
.59377 

•59453 
•  59530 

70 
76 
76 
76 
76 

9.91034 
.91049 
.91065 
.91080 
.91095 

15 
i5 
15 
i5 

10.63961 
.64043 
.64125 
.  64207 
.64289 

82 
82 
82 
8a 

15 

16 

17 
18 

19 

10 

20 
30 
4° 
5° 

12.4 

27-6 
41.5 

55-3 
69.1 

12.3 

27-3 
41.0 

54-6 
68.3 

12.  I 

27.0 
40-5 

54-0 
67.5 

20 

21 

22 
23 
24 

9.90183 
.90204 

.90235 
.90256 

Is 
i5 

_  a 

10.59605 
•59683 
.59760 

.59837 
•  599H 

7o 
77 
77 

9.91110 
.91126 
.91141 

•9"56 
.91171 

15 

10.6437! 
.64453 
.64536 

.  646  i  § 
.  64701 

82 
82 
82 
83 

20 

21 

22 
23 
24 

6 

80 

8.0 

9-1 

79 

11 

78 

7-8 
9.1 

3 

27 
28 
29 

9  .  90266 
.90281 
.90297 
.90312 
.90328 

15 

a 

10.59991 
.60068 
.60145 
.60223 
.60306 

77 
77 
7? 
7? 
7? 

_£ 

9.91187 
.91202 
.91217 
.91232 
•9124? 

15 
15 

15 

10.64784 
.64867 
.64950 

•65033 
.65116 

83 
83 
83 
83 

25 
26 

27 
28 

29 

9 

10 

20 

40 
50 

12.0 

'3-f 
26-6 
40.0 

53-3 

a 

39.5 
52.6 
65.8 

11.7 
13.0 
26.0 
39-0 
52.0 
65.0 

30 

32 
33 
34 

9.90343 
.90359 
.90374 
.90389 
.90405 

15 
15 

iS 

15 

10.60378 
.60455 
•60533 
.60611 
.6o68§ 

77 

78 
7? 

9.91263 
.91278 
.91293 
.91308 
.91323 

15 
i-5 

15 

10.65199 
•65283 

•65365 
.65450 

•65534 

83 
83 
83 
84 

30 

32 
33 
34 

6 
7 
8 

77 

7-7 

9.0 

IO.2 

76 
tj 

10.  I 

75 

10.  0 

35 
36 

39 

9  .  90420 
.90436 
•90451 
•90467 
.90482 

^5 

10.60765 

.60844 
.60923 
.61001 
.61079 

78 
78 

9.91338 
•9^354 
.91369 

.91384 
.91399 

15 
15 
15 

10.6561? 
.6570! 

.65785 
.65870 

.65954 

84 
84 
84 
84 

OT 

11 

37 
38 
39 

9 

10 

20 
3° 
4° 
5° 

"'I 

i*  8 
25  6 
38.5 

£:! 

11.4 
ia.j| 

25-3 
38.0 

63-3 

12.5 
25.0 

37.5 

40 

42 
43 
44 

9-90497 
•90513 
•90528 
•90544 
•90559 

i5 

10.61158 
.61235 
.61315 

.61393 
.61472 

1 

9.91414 
.91429 

.91445 
.91460 

.91475 

15 

10.66038 
.66123 
.6620? 
.66292 
.66377 

54 

84 
84 
84 

85 

40 

42 
43 
44 

6 
7 
8 
9 

6 

o. 

0. 

o. 
o 

5 
3 
5 

i 

45 
46 
47 
48 

49 

9.90574 
.90590 
.90605 
.90621 
•90635 

15 
15 

10.61551 
.61630 
.61709 
.61783 
.6186? 

78 
79 
79 
79 
79 

f.f! 

9.91490 
.91505 
.91526 

•9'535 
.91550 

15 

10.66462 
.66547 
.66632 
.6671? 
.66803 

85 

85 

46 
47 
48 

49 

10 

20 
30 
4° 
5° 

0. 

o. 

0. 

o. 
o. 

i 

2 

3 
* 

50 

51 

52 
53 
54 

9.90651 
.90667 
.90682 
.90697 
.90713 

15 
15 

.  a 

10.61947 
.62026 
.62105 
.62185 
.62265 

79 

1  79 
1  79 

'£ 

9.91565 
.91581 
.91596 

.91626 

!5 
15 

!  J5 

10.66883 
•66974 
•67059 

.67231 

Qf. 

50 

52 
53 

54 

6 

I 

9 

16 

1.6 

2.  1 
2.4 

LS 

1.8 

2.5 
2-3 

15 

2.O 
2.2 

57 
58 
59 

9.90728 
.90744 
.90759 
.90774 
.90790 

jS 

T  r 

10.62345 

.62504 
.62585 
.62665 

7Q 
80 
86 

1  Sn 

9.91641 
.91656 

'91685 
.91701 

15 

T5 
15 

15 

I  r 

10.6731? 

.67403 
.67490 

•67576 
.67663 

86 
85 

8£ 
8£ 

55 
56 

% 

59 

10 

20 
30 
40 
50 

5-3 
8.0 

2.6 

12.9 

2-5 

5-° 

7-5 

TO.O 

12.5 

00 

9.90805 

1  5 

10.62745 

9.9I7I6 

1  J 

10.67749 

<>0 

' 

LOST.  Vers. 

It 

Loer.  Exsec 

n 

Lou.  Vers. 

It 

I,  OK.  Ex  see. 

it 

| 

\ 

p. 

433 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL    SECANTS. 

8O°  81° 


/ 

Log.  Vers. 

D 

Log.  Kxsec. 

J> 

Log.  Vers. 

Log.  Exsec. 

> 

P.  P. 

0 

I 
2 

3 

4 

9.9I7I6 
.9I73? 
•91746 
.9176! 
-91776 

15 
15 
15 

10.67749 
.67836 
.67923 
.68010 
.68097 

87 

87 
87 

9.92612 
.92625 
.9264! 
.92656 
.92671 

15 
14 
15 

10.73173 

.73273 
.73368 
.73463 
•73558 

95 
94 
95 
95 

0 

I 

2 

3 
4 

6 

90 

9.0 

80 

8.0 

I 

7 
8 

9 

9.9179! 
.91807 
.91822 

.91837 
.91852 

r§ 

15 

15 

10.68184 
.68272 

.68359 
.68447 
.68534 

8? 
8? 

8? 

00 

9.92686 
.92706 
.92715 
.92730 
.92745 

15 
14 

15 
14 

15 

10.73653 

•73748 
.73844 
•73940 
.74035 

95 
95 

96 

95 

r\f\ 

6 

8 
9 

8 

9 
10 
20 

3° 
40 

5° 

12.0 

15-0 
30.0 

45-o 
60.0 
75-° 

10.  g 

12.0 

26.g 
40.0 

53  3 
66.5 

10 

ii 

12 

14 

9.91867 
.91882 
.91897 
.91912 
.91927 

15 
15 

10.68622 
.68716 

.68798 
.68885 
.68975 

88 
88 

oo 
oo 

9.92759 
.92774 
.92789 
.  92804 
.92813 

15 

15 

10.7413! 
.7422? 

.74324 
.74426 

-745'7 

90 

96 
96 
96 

10 

ii 

1  2 

13 

14 

6 
7 

9 

0.9 

i  .6 

8 

0.8 

!! 

17 
18 

'9 

9.91942 

.91972 
.91987 
.  92002 

15 

15 
15 

10.69063 
.69152 
.69246 
.69329 
.69413 

89 

.92848 
.92862 
.9287? 
.92892 

ii 

10.74613 
.74716 
.7480? 
.74905 
.75002 

96 
97 
97 
91 
97 

r»*7 

l| 

17 

18 

9 
10 

20 

3° 
40 

5° 

1.2 

i-3 

3-o 
4-5 
6.0 

7-5 

I.O 
1.2 

2"  6 

4.0 

3:1 

20 

21 

22 

23 
24 

9.92015 
.9203! 

•  92046 
.9206! 
.92075 

15 
15 
15 

10.6950? 

•69596 
.69686 

.69775 
.69865 

89 
89 
89 
89 

9.92907 
.9292! 
.02936 
.92951 
.92965 

[| 

H 
i-5 

10.75099 

•75295 
•75393 
•7549' 

97 

98 

97 
98 
98 

20 

21 
22 
23 
24 

6 
7 
8 

7 

0.7 

0.8 
0.6 

6 

0.6 
0.7 
0.8 

25 
26 
27 
28 
29 

9.92091 
.92105 
.9212! 

.92136 
.92151 

15 
15 
15 
14 

10.69955 
.70044 

.70134 
.  70224 

-70315 

9° 

89 
90 

90 

90 

9.92986 
.92995 
.93009 
.93024 
.93039 

15 
14 

10.75589 
.75688 
•75785 
•75885 
-75984 

98 
98 
98 
99 
99 

25 
26 
27 
28 
29 

9 
10 

20 
3O 
40 

5° 

1.6 
i  .1 

2-3 

3.5 

4-6 
5-8 

0-9 
i  .0 

2.O 
4-0 

5-0 

30 

32 
33 

34 

9.92166 
.92181 
.92196 
.92211 
.92226 

'5 

15 
15 

10.70405 

-  70495 
.70585 
.70677 
.70768 

9° 

96 

91 
90 
91 

9-93053 
.93068 
.93083 
.9309? 
.93112 

15 
14 

10.76083 
.76182 
.76282 
.76382 
.76481 

99 
99 
99 

IOO 

99 

30 

32 
33 
34 

6 

1 

5 

0.5 

0.6 

4 

0.4 
0.4 

35 
36 

% 

39 

9.92240 
.92255 
.92276 
.92285 
.92306 

15 
15 

10.70859 
.70950 
.71041 

.7H33 

.71224 

91 
91 
9i 

9! 
9! 

_  f 

9.93127 
•93141 
•93156 
.93171 
.93185 

14 
H 
H 
15 
H 

T  7 

10.7658! 
.7668! 
.76782 
.76882 
.76983 

IOO 
IOO 
IOO 
100 

35 

36 
37 
38 
39 

9 

10 

20 

40 
5° 

0.7 
0-8 
J-6 

3-2 

4.1 

06 

o.g 

2.0 
2.6 
3-3 

40 

41 
42 
43 

44 

9.92315 
.92330 
•92345 
.92360 

.92374 

15 
15 

10.71316 
.71408 
.71500 
.71592 
.71684 

91 
92 
92 
92 
92 

9.93200 
.93214 

.93244 
•93258 

14 

14 
15 

14 
14 

10.77083 
•77184 
.77286 
.77387 
-77488 

IOO 
101 

10! 

101 

10! 

40 

42 
43 
44 

6 

I 

9 

I«5 

1.8 

2.6 

2.3 

15 

1  § 
'•7 

3.0 
2.2 

45 
46 
47 
48 

49 

9.9238§ 
.92404 
.92419 
.92434 
.92449 

15 

15 
15 

10.71775 
.71869 
.7196! 
.72054 

.72147 

92 
92 
92 
93 
92 
_  ~ 

9.93273 
.9328? 
.93302 
.93317 
.93331 

a 
15 
H 

10.77590 
.  77692 
•77794 
•77896 
•77998 

102 
102 
102 

4I 
46 

47 
48 

49 

10 

20 

3° 
40 

50 

2.6 

10.  § 
12.9 

2-5 

5-0 

7-5 

IO.O 

".5 

50 

5' 

52 
53 

54 

9.92463 
•92478 
•92493 
.92508 
.92523 

15 
14 

15 

10.72246 
.72333 
.72427 
.72526 

.72614 

93 
93 
93 
93 
93 

r\3 

9.93346 
.93366 

•93375 
•93389 
•  93404 

H 
15 

10.78101 
.78203 

•78306 
.78409 

•78513 

I  O2 

103 
103 
103 

50 

52 

53 
54 

6 

8 
9     : 

[4 

2.2 

55 
56 

11 

59 

9.92538 
.92552 
.9256? 
.92582 
•92597 

15 

I  c 

10.7270? 
.7280! 
•72895 
.72990 
.73084 

93 
94 
94 
94 
94 
ol 

9.93419 
.93433 
.93448 
.  93462 
•93477 

H 

i4 

H 

H 

H~ 

10.78616 
.78720 
•78823 
.7892? 
•79031 

103 
104 
103 
104 

IOA 

P 

57 
58 
59 

10 

20       < 

30 

40     c 

50      1'. 

>-4 

'  .2 

>-6 
.1 

00 

9.92612 

1  J 

10.73173 

9.93491 

10.79136 

60 

Log.  \  ers. 

7> 

.oar.  Kxsec. 

Loir.  \>rs. 

j> 

' 

P.   P. 

434 


TABLE   VIII.— LOGARITHMIC    VERSED    SINES    AND    EXTERNAL    SECANTS. 

82°  83° 


/ 

LOST.  Vers. 

D 

Log.  Exsec. 

y>  \ 

Log.  Vers. 

I) 

MS.  Exsec. 

] 

P.  P. 

0 

9-93491 

10.79136 

I 

9-94356 

10.85765 

T  I? 

0 

I 

2 

3 

.93506 
•93526 
•93535 

*4 
14 

a 

.  79240 

•79345 
.79450 

105 
104 

,  .94370 
.94384 

•94398 

H 
H 

.85884 
.86001 
.86119 

1  1  / 
II? 

II? 

T  l8 

I 

2 

3 

4 

•93549 

J4 

•79555 

105 

.94413 

*4 

T  A 

.86237 

T  TO 

4 

I 

8 

9.93564 
•93578 
•93593 

14 
H 

a 

10.79666 
.79766 
•79871 
•7997? 

I05 
105 
105 

106 

Tr»A 

9.9442? 
•94441 
.94456 
•94470 

14 

H 

10.86355 

.86474 
.86592 
.86711 

§ 

"8 

119 

T  Tri 

6 

7 
8 

6   i 
7 
8 
9 

10 

3° 

3.0 

5-i 
'7-3 
9-5 

ZI.fi 

12.0 
14.0 

16.0 
18.0 
20.  o 

9 

.93622 

'i 

.80083 

.94484 

14 

.86831 

119 

T  TO 

9 

20 

40.0 

10 

ii 

12 
13 

9.93636 
•93651 
.93665 
.  93680 

i* 

14 
a 

10.80189 
.  80296 
.  80402 
.  80509 

106 

105 
107 

9-94498 
.94512 
.94527 
.94541 

14 

14 

14 
14 

10.86956 
.87076 
.87196 
.87316 

119 

120 
120 
120 

10 

ii 

12 
13 

40 

5°  x 

38  ".3 

80  o 

100.0 

14 

•93694 

.80615 

107 

•94555 

14 

.87431 

14 

15 

16 

17 
18 

19 

9.93709 
.93723 
•93738 
.93752 
.93767 

14 

a 

H 

U 

10.80723 
.  8083  i 
•80938 
.81045 
•81154 

107 
!of 

1  08 
1  08 

Tr.Q 

9.94569 
.94584 
•94598 
.94612 
.94625 

T4 
14* 

it 
H 

10.87552 
.87673 
.87794 
.87916 
.88038 

121 

12! 
12! 
!I22 

TOO 

1! 

17 
18 

19 

6 

7 
8 
9 

10 

110 

II.  0 
12.  § 

14-6 

16. 
18.= 

IOO 

10.  0 

ii.  6 

13-3 
15.0 

20 

21 

22 

9.93781 
.93796 
.93816 

10.81262 
.81371 
.81479 

io§ 
io§ 

9.94646 

•94655 
.94669 

*4 
H 

I0.88l6o 
.88282 
.88405 

122 
122 

20 

21 

22 

20 
3° 
40 
50 

36-6 
55-° 
733 
91-6 

33-3 
50.0 
66.6 
83-3 

23 

24 

.93824 
.93839 

,  ? 

.81588 
.8169? 

109 
109 

.94683 
.9469? 

14 

.88528 
.88651 

123 
123 

TO  J 

23 

24 

25 
26 
27 
28 

!  93868 
.93882 
•93897 

14 
H 
H 

10.81805 
.81916 
.82025 
.82135 

109 
109 
109 
no 

9-947II 
.94726 
.94740 
•94754 

4 

10.88775 
.88898 
.  89022 
.89147 

124 
123 
124 

124 

11 

27 
28 

6 

7 

3 

2 

0.2 
O.2 

29 

•93911 

.82245 

T  fA 

.94768 

.' 

.89271 

124 

29 

8 

0.4 

0.2 

30 

9-93925 

Jf 

10.82356 

I  IO 

T  TO 

9.94782 

10.89395 

125 

30 

IO 

0.5 

32 
33 

•93940 

•93954 
.93969 

14 
* 

.82465 
.82577 
.82688 

no 
III 

•94796 
.94816 
.94825 

• 
H 

.89521 
.89647 
.89773 

125 

125 

126 

TOA 

32 
33 

20 

3° 
4° 
50 

I.O 

2.0 

2-5 

I.O 

1:1 

34 

•93983 

7, 

.82799 

T  T  ? 

.94839 

u 

*4 

.89899 

34 

3 

9-9399? 
.94012 

14 
H 

10.82916 
.83022 

III 

III 

T  I? 

9  :  9486? 

14 

14 

10.90025 
.90152 

126 
126 

P 

37 
38 
39 

.94025 

.94041 

•  9405=; 

14 
14 

.83133 
.83245 
.833^ 

112 
112 

.9488? 

.94895 
.94909 

H 
H 

.90279 
.  90406 
.90533 

127 
127 
12? 

TO0. 

IK 

39 

6 

7 

I 

O.I 
O.I 

6 

0.0 

o.o 

40 

9.94o6§ 
.94084 

H 

10.83470 
.83583 

T  1  0 

112 
112 
T  T7 

9-94923 
.94938 

14 
H 

•t  A 

10.9066! 
.90789 

I2o 

12? 

40 

8 
9 

10 

O.  I 
O.I 
O.I 

o.o 

O.I 
O.I 

42 

.94098 

•4 

•83695 

.94952 

14 

.9091? 

42 

20 

0.3 

O.I 

o  5 

43 

.94112 

14 
*? 

.83809 

Ir3 

.94966 

•4 

.91045 

129 

43 

40 

o-fi1 

44 

.94127 

*4 

.83922 

JI? 

.94980 

• 

.91175 

129 

44 

50 

o  8 

0.4 

45 

9.94141 

10.84035 

H3 

9-94994 

14 

10.91304 

129 

45 

46 

.94155 

.84149 

114 

•95008 

.9H34 

130 

46 

47 
48 

.94170 
.94184 

l| 

.84263 
.8437? 

114 
114 

.95022 
•95°36 

*4 

.91564 
.91694 

136 

47 
48 

49 

•94198 

14 

.84492 

114 

.95050 

.91825 

130 

49 

6 

50 

52 

9.94213 
.94227 
.94241 

H 

10.84607 
.84721 
.84837 

114 

"5 

9.95064 
-95078 
•95093 

H 
14 

10.91956 
.  92087 
.92218 

131 
131 
171 

50 

52 

7 
8 
9 

IO 

.7 

9 

.2 

•4 

4 

i  8 

53 

54 

.94256 
•94270 

14 

•84952 
.85068 

116 

T  T  ? 

.95107 
.95121 

H 

.92350 

.92482 

1  Jl 
132 

53 
54 

20 
30 

4° 

•I 

.2 

9-6 

4-6 
7.0 

9-3 

55 

9.94284: 

T  ? 

10.85183 

TI5 

T  lf\ 

9.95I35 

H 

10.92614 

132 

55 

5° 

12.  1 

11.6 

56 

57 

.94299 
.94313 

14 
14 

.85299 
.85416 

U6 

•95149 
•95l63 

.9274? 
.92886 

133 

56 
57 

58 

•9432? 

.85532 

.95177 

14 

.93014 

133 

58 

59 

•94341 

14. 

.85649 

117 

117 

.95191 

•9314? 

I  "34 

59 

60 

9-94356 

10.85765 

9.95205 

10.9328! 

60 

' 

Log.  Vers. 

LOST.  KXSPC. 

D 

Log.  Vers. 

n 

Loir.  Exsec. 

D 

' 

F.  1» 

TABLE   VIII.— LOGARITHMIC   VERSED    SINES    AND    EXTERNAL   SECANTS. 

84°  85° 


; 

Log.  Vers. 

D 

Log.  Exsec.  -D 

Log.  Vers. 

J> 

Log.  Exsec. 

jj 

• 

P.  P. 

0 

I 

2 

3 

4 

9.95205 
.95219 
.95233 
.9524? 
.9526! 

14 

H 

H 
H 
H 

H 
H 

H 
H 
H 

'! 
H 
14 

H 

14 
14 

H 

H 
13 
H 
H 

H 

H 
13 

H 

n 

10.9328! 
.93416 

•93551 
.93686 
.9382! 

134s 

135 

135 

I3i 

I36 

136 
137 
137 

^38 
138 
139 

140 
146 
146 

i4t 
141 
142 

u 

143 
143 
144 
144 
144 

145 
145 
145 

146 
146 

147 

148 
149 
149 

150 
150 
151 

151 

152 
152 
153 

154 
'55 

'5§ 

i57 
i5? 

9.96039 

.96053 
.96067 
.96081 
.96095 

H 

H 

H 
13 

14 

H 
13 

H 

14 
13 
H 

H 
13 

14 
'3 

H 
13 

H 
13 
13 

14 
13 
13 

14 

\l 

13 

i! 

H 

13 

'3 
13 

II  .02010 
.02l6§ 
.O232? 
.02487 
.  02645 

158 
159 
'59 

1  66 
161 
16! 
16! 
162 
163 

164 
165 

|65 

167 
167 
1  6? 

1  68 
169 

169 
170 

171 

172 
173 
173 

174 
174 

175 
176 
176 
177 

1  78 
179 
179 
1  80 
1  86 
18! 

182 

184 
185 
185 
1  86 

1  88 
189 
189 
190 

193 

i 

2 

3 

4 

6 
7 
8 

9 

10 

20 

3° 
40 
50 

6 
7 
8 
9 

10 

20 

3° 
40 
50 

6 
7 
8 
9 
10 
20 
3° 
4° 
50 

6 
7 
8 
9 

10 

20 

30 
40 
5° 

6 

87 

9 

10 

20 
3° 
4° 
50 

6 

I 

9 

10 

20 

3° 
40 

5°  i 

190 

19.0 

22.  I 
25-3 

5:1 

63.3 
95.0 
126.6 
158.3 

170 

17.0 

22.g 
25'5 

56.6 
85.0 

"3-3 
141.6 

ISO 

15-0 
17.5 
20.  o 
22.5 
25.0 
50.0 

100.  o 
125.0 

130 

13.0 

'9-5 

21-6 

43-3 
65.0 
86.g 
io8.I 

7 

0.7 

0.8 
0.9  ( 
i  .6  ( 
i.i 

2.3  • 

3-5  . 

4-6  < 
5-8  . 

H 

•9   ] 

I, 

i 

2 

2 
2 
3 

6 

9 

12 

15 

I 

I 
1 
2 
2 

5 

8 

10 

*3 

I 
i 

i 

i 

2 
•2 

4 
7 
9 
ii 

9 

0.9 

I.O 
I  .2 

i-3 

I.J 

3-° 

11 

7-5 

6 

j.6 
5-7 

3.8 

>«9 

c.o 

>.O 

)X> 

.0 

.0 

14 

3o 

?.o 
t.o 
t-o 

J.O 
D.O 
3.O 
5.0 
3.O 
3.0 

60 

5.o 
B-6 
t-3 
*'° 

3-3 

D.O 

5.6 
3-3 

10 

J1 

3-6 
i  .0 

5.6 

D.O 

3-3 
5.6 

8 

0.8 
0.9 
i  .0 

I  .2 

2-6 
4.0 

5 

0.5 

0.6 
0.6 

o'" 

2-5 

3-3 
4.1 

13 

1.6 
1.8 

1 

7 
8 

9 

9.95275 

.95303 
•9531? 
•95331 

10.93957 
.94093 

•94366 
.94503 

.96122 
.96136 
.96150 
.96163 

II  .02807 
.02968 
.03I2§ 
.03291 
-03453 

6 

8 
9 

10 

ii 

12 
13 
H 

9-95345 
•95359 
•95373 
.95387 
.95401 

10.94641 

•94778 
.94917 
.95055 
.95194 

9.96I7? 
.96191 
.96205 
.96213 
.96232 

II  .0^615 
.03780 

.03944 
.  O4  I  08 

.04273 

10 

ii 

12 
13 

H 

11 

17 

18 
19 

9.95415 
.95429 

.95443 
•95457 
•95471 

10.95333 
•95473 

•95753 
.95894 

9.96246 
.96259 

.96273 
.96287 

.96301 

n.04438 
.  04604 
.04771 

•  .04938 
.05106 

16 

17 
18 

19 
20 

21 

22 
23 

24 

20 

21 

22 

23 
24 

9.95485 
•95499 
.95513 
.95527 
•95540 

10.96035 
.96176 
.96318 
.96461 
.96603 

9.96314 
.96328 
.96342 
.96355 

11.05274 

•05443 
.05612 
.05782 
.05952 

25 
26 
27 
28 
29 

9-95554 
.95568 
.95582 

•95596 
.95616 

10.96745 
.96889 
.97033 
.97177 
.97322 

.96397 
.96416 
.  .96424 
.96438 

ii  .06123 
.06295 
.  06467 
.  06640 
.06813 

% 

27 
28 

29 

30 

32 
33 
34 

9.95624 
.95638 

'.95666 
.95680 

10.97467 
.97612 
.97758 
.97904 
.98056 

9.9645! 
.96465 

.96479 
.  96492 
.96506 

ii  .06987 
.0716! 

•07336 
.07512 

.07688 

30 

31 

32 
33 
34 

P 

37 
38 
39 

9-95693 
•9570? 
.9572! 
•95735 
•95749 

10.9819? 

.98345 
.98492 
.  98646 
.98789 

9.96519 
.96533 
.96547 
.96566 
.96574 

ii  .07865 
.  08043 
.08221 
.  08400 
.08579 

1 

39 

40 

42 
43 

44 

•95777 
•95791 
.95804 

.95818 

10.98933 
.9908? 
.9923? 
•9938? 
•99538 

9.96588 
.  9660! 

•9~'5 
.  96629 

.96642 

11.08759 
.08940 
.09121 
.09303 
.09486 

40 

42 
43 
44 

45 
46 
47 
48 
49 

9-95832 
.95845 
.95860 

.95874 
.95888 

10.99689 
.99841 
10.99993 
11.00145 
.00298 

9.96656 
.96669 
.96683 
.96697 
.96716 

i  i  .  09669 

.09853 
.  10038 
.  10223 
.  10409 

45 
46 

47 
48 

49 

50 

52 
53 

54 

9-95901 
•959^5 
.95929 
•95943 
•95957 

ii  .0045! 
.00605 
.00759 
.00914 
.01069 

9.96724 
.9673? 
.96751 
.96764 
.96778 

11.10595 
.  10783 
.10971 
.11160 

•II349 

50 

52 
53 
54 

55 
56 
57 
58 
59 

9.95970 
.95984: 

•95998 
.96012 
.  96026 

ii  .01225 
.01381 

.0153? 
.01694! 
.01852 

9.96792 
.96805 
.96819 
.96832 
.  96846 

11.11539 
.11736 
.11922 
.12114 

.1230? 

P 

57 

58 
59 

.4   2.3 

'§   4'6 
7.2   7.0 

9-6   9-3 

2.1   II  .6 

2.2 

J:l 

9.0 

II.  2 

60 

9.96039 

II  .02010 

9.96859 

ii  .  12501 

00 

'    Log.  Vers. 

.Off.  Exsec.   7>   Loar.  Vers.  J> 

436 


TABLE    VIII.— LOGARITHMIC   VERSED    SINES    AND    LXTERNAL   SECANTS. 

86°  87° 


; 

Loe.  Yers. 

D 

,og.  Kxsec. 

D 

Log.  Yers. 

1) 

Lou:.  Kxsec. 

i 

0 

i 

2 

9.96859 

.96873 
.  96887 

13 

14 

II  .  I250I 
.12696 
.12891 

195 
195 

.97679 
.97692 

<co  cow 

11.25785! 
.  26046 
.  26297 

255 

256 

0 

i 

2 

3 

.96906 

13 

1  7 

.13087 

196 

.97705 

*J 

•26554 

257 

•7  Cn 

3 

250 

240 

4 

.96914 

,  ft 

.13284 

Tr.o 

.97718 

-  ft 

.26814 

259 

4 

6 

7 

25.0 
29.1 

24.0 
28.0 

I 

7 

.96941 
•96954 

•Occoccocc 

II  .1^482 
.13680 
•13879 

195 

198 
199 

9-97732 
•97745 
•97758 

•O  coccocc 

11.27074 

•27336 
•27599 

262 
263 

6 

7 

8 
9 
10 

20 

33-3 
37-5 
4I-6 
83-3 

32  0 

36.0 
40.0 

80.0 

8 
9 

.96968 
.96981 

3 

.  U079 
.  14286 

201 

•97772 
.97785 

»J 

13 

T  5 

.  27864 
.28131 

266 

8 
9 

30 
40 
50 

125.0 
i66.f 
208.3 

I2O.O 
l6o.O 
200.0 

10 

ii 

9.96995 

•97008 

'3 

13 

II.  14482 
.  14684 

202 

9-97798 
.97811 

13 
13 

11.28398 
.28668 

26§ 

10 

ii 

12 
13 
14 

.97022 
•97035 
•97049 

i] 

.1488? 
.15092 
.15297 

203 
204 
205 

•97825 
•97838 
•97851 

13 

.28933 
.29211 

•29485 

270 
272 

274 

i-  3 

12 
13 

14 

6 
7 

230 

23.0 

26.8 

220 

22.  O 

25-6 

15 

16 
17 

9.97062 
.97076 
•  97o8§ 

13 

I  5 

II.  15502 
.15709 
.15917 

206 
208 

•70°, 

9.97864 
.97878 
.97891 

•O(CO  COCC 

i  i  .  29766 

.30037 
.30316 

275 
277 

278 

15 

16 

17 

9 

10 

20 
3° 

3°  6 
Tfiij 

29.3 
33.0 

73-3 

I  IO.O 

18 

19 

.97103 
•97II6 

13 

I  3 

.  l6l25 
•16334 

20§ 

.97904 
•9791? 

»J 

13 

•30596 
•30878 

279 
282 

/•>QS 

18 
19 

40 
50 

'53-3 
191.6 

146.6 
183-3 

20 

9.97130 

13 

11.16544 

9-9793' 

ii  .31162 

283 

•7  pa 

20 

21 

.97H3 

X 

.16755 

•97944 

13 

•3*44? 

265 
0Q- 

21 

22 

.97157 

.1696? 

•9795? 

I3 

.31734 

*4 

22 

210 

200 

23 

24 

.97170 
•97183 

13 

.17186 

•17394 

213 
214 

tl  A 

.97970 
.97984 

13 

.32023 
•32313 

258 
296 

23 

24 

6 

I 

21.  C 

24-5 

28.  c 

20.  o 
23-3 
26.6 

25 
26 

9.97197 
.97216 

T  3 

II.I7609 
.17824 

214 

2,| 

9-97997 
.98016 

13 

ii  .32606 
.32900 

294 

25 
26 

9 

10 
20 

35-c 
70.  c 

30.0 
33-3 
66.6 

27 

28 
29 

•97224 
.9723? 
.97251 

OCCO  (CO  « 

.18041 
.18259 
.1847? 

215 

218 

218 
~,p. 

.98023 
.98035 
.98050 

I3 

13 

13 

•33196 
•33494 
•33793 

298 
299 

0/-VT 

27 
28 

29 

0° 

40 

50 

105.0 
140.0 

IOO.O 

166.6 

30 

9.97264 

13 

1  1  .  1  8697 

219 

9.98063 

13 

11.34095 

301 

30 

32 
33 
34 

•9727? 
.97291 

.97304 
.97318 

OCCO(COCCO  < 

•I9I38 
.19584 

221 
222 
223 

•98075 
.98089 
.98102 
.98116 

O  CO  co<CO 

•34398 
.34704 
.35011 

•35321 

3°3 
305 
3<>? 
309 

*»  1  f 

32 
33 
34 

6 
7 
8 

19.0 

22.1 
25-3 

4  3 

0.4  0.3 
0.4  o.§ 
o.§  0.4 

35 

9-97331 

1  j 
Ts 

II.  19809 

224 

-7-71? 

9.98129 

»3 

11.35632 

3*i 

35 

9 
10 

31-6 

36 
37 

•97345 
.97358 

13 

13 

.20034 
.  2026T 

22-, 
227 

.98142 
.98155 

13 

.35946 
.36261 

3i5 

36 
37 

20 
30 

4° 

95-o 
126-6 

I.§   I.O 
2.0   1-5 
2.g   2.0 

|3« 
!  39 

•97371 
•97385 

Ta 

.  20489 
.20717 

228 

.98163 
.98181 

!3 

13 

.36579 
•36899 

320 

38 

39 

50 

"SB.! 

3*3  2.5 

40 

9-97398 

I3 
f  3 

I  I  .  20947 

230 

9.98195 

J3 

11.37221 

322 

40 

41 

•97412 

13 
13 

.21178 

.98208 

T-2 

.37546 

324 

41 

2 

I   6 

42 

.97425 

.21410 

.98221 

.37872 

100 

42 

6 

O.2 

0.    0.0 

43 

44 

•97438 
•97452 

13 

.21643 
.21877 

233 
234 

.98234 
•9824? 

13 

.38201 
.  -38532 

328 
33? 

•»_2 

43 
44 

9 

O.2 
O.2 
0-3 

o.   o.o 
o.   0.6 

0.    O.I 

41 
46 

9.97465 
•97478 

13 

II  .22112 
.22349 

235 

236 

9.98266 
•98273 

13 

1  5 

i  i  .  38866 
.39201 

333 
335 

45 
46 

10 

20 
30 

0.6 

1.0 

o.   o.i 
0.3  o.i 
0.5  0.3 

47 

48 

•97492 
.97505 

13 

•22585 
.2282§ 

239 

.98287 
.98300 

13 

.39540 
.39886 

338 
340 

•2/1  5 

47 
48 

4° 
50 

!:i 

o.£  0.3 
0-8  0.4 

49 

•97519 

•23065 

239 

*•>  i  T 

•98313 

J3 

.40224 

343 

„  .  P 

49 

50 
51 

52 

9-97532 
•97545 
•97559 

13 
13 

•23548 

.  23792 

241 

242 

243 

9.98326 
•98339 
.98352 

•occo  co  f 

ii  .40569 
.40918 
.41269 

rvoo  I-H  <( 

•i-  -3-  ISM 
O  CO  CO  f 

50 

52 

6 

«4 

1.4 

13   13 

53 

54 

.97572 
•97585 

.24037 

.24283 

246 

.98365 
.98378 

•J 

13 

.41622 
.41979 

J3J 

356 

53 

54 

7 
8 
9 

2.1 

1.6   1.5 
1.8   1.7 

2.0    I.g 

H 

9-97599 
.97612 

13 

11.24530 

.24778 

247 

248 

9.98392 
.  98405 

'3 

13 

11.42338 
.42699 

359 

| 

10 

20 
30 

4-6 
7-o 

2.2    2.1 

4-5   4-3 

6.7   6.5 

H 
59 

.97625 

•97639 
.97652 

13 

13 
13 

.25028 
•25279 
•25531 

250 
251 

252 

2s4. 

.98418 
.98431 
.98444 

O  CO  CO<C< 

.43064 

•43431 
.43802 

367 
37o 

59 

4° 

5° 

9-3 

9.0   8.6 

1.2   10.  § 

60 

9-97665 

11.25785 

9-984=;? 

11.44175 

60 

i  ' 

Log.  Yew. 

y> 

Log.  Kxser. 

/>  1 

Los.  Yers. 

5 

L.OST.  Kxsec. 

j> 

' 

P.  J 

p. 

437 


TABLE   VIII.— LOGARITHMIC   VERSED    SINES   AND    EXTERNAL    SECANTS. 

88°  89° 


/ 

Log.  Vers. 

J> 

Log.  Exsec.   D 

Log.  Vers. 

It 

Log.  Exsec. 

J> 

/ 

P.  P. 

0 

2 

3 
4 

9.9845? 
.98476 
.98483 
.98495 
.98509 

13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 
13 

\l 

13 
13 
13 
13 
13 
13 
13 

13 
12 

13 
13 
13 
13 
13 
13 
13 
12 

13 
13 
13 
13 
12 

13 
13 
13 
13 
12 

13 
13 

I3 

12 
13 
13 

11.44175 
.44551 

•44931 
.45313 
.45699 

376 

379 
382 
386 
389 
392 
395 
399 
402 

406 
409 
413 
417 
426 

425 
428 
432 
436 
446 

445 
449 
454 
458 
463 
46? 
472 
47? 
482 

48? 
492 
498 
504 
509 

515 

526 
527 
533 
539 
545 
552 
559 
566 

573 
58i 
588 

595 
604 
611 
626 

623 
638 

646 
656 
666 

675 
685 

696 
7o? 
719 
730 

9.99235 
.99248 
.99261 
.99274 
.99287 

12 
13 
13 
13 
12 
13 

I3 

12 
13 
12 
13 
13 
12 

13 

13 
12 

13 
12 

13 
12 
13 
13 
12 

13 
12 

13 

12 

13 

12 

1-3 

12 

13 
12 

13 
12 
12 

13 
12 

13 
12 

13 
12 
12 
13 

12 

13 
12 
12 
13 
12 
12 

13 
12 
12 

13 
12 
12 
12 

11 

11.75050 
.75792 
.7654? 
.77316 
.7809? 

742 

755 
768 
781 

795 
809 
825 
840 
856 
872 
896 
908 
927 
947 
967 
989 
1009 
1034 
1059 
1085 

III2 
1140 
II7I 
1203 
12^6 
1271 
1309 

1349 
1391 

H36 
1485 

1537 
1592 
1652 
1716 

1785 
1861 

1943 
2033 

2131 

2246 
2361 
2495 
2645 
2815 
3009 
3231 
3489 
3791 
4152 

4588 
512? 
5812 

670? 
,_,,*• 

0 

i 

2 

3 
4 

13  13 

6   i.  §    .3 
7   1.6    .5 
8   1.8    -7 
9   2.0    .9 

JO    2.2      .1 

20   4.5   4-3 

30    6.7    6.5 
40    9.0    8.5 

50  ii.  3  10.  § 

12 

6      .5 

I   :l 

g      .9 

IO         .  I 
20         .1 

30     6.2 

40   8.3 

50    10.4 

6 

8 
9 

9.98522 

.98535 
.98543 
.98562 
.98575 

11.46088 
.  46486 
.46876 

.47275 
.4767? 

9.99299 
.99312 

.99325 
.99338 
•99351 

11.78892 
.79702 
.80527 
.81367 
.82223 

6 

8 
9 

10 

ii 

12 
13 
14 

9.98588 
.98601 
.98614 
.98627 
.  98640 

11.48083 

.48493 
.48906 

.49323 
•49743 

9-99363 

•99376 
.99389 
.99402 

.99415 

1  1  .  83095 
.83986 
.84894 
.85821 
.86768 

10 

ii 

12 
13 

H 

15 

16 

17 
18 

19 

9.98653 
.98666 
.98679 
.98692 
.98705 

11.50168 
.50597 
.51029 
.51466 
•51905 

9.99428 
.99446 

•99453 
.99466 

•99479 

11.87735 
.88724 

•89735 
.90769 
.91829 

15 

16 

17 
18 

19 

20 

21 

22 
23 
24 

9.98718 
.98731 
.98744 
.98757 
.98770 

11.5235! 
.52801 
.53255 
.53713 
.54176 

9.99491 
.99504 

•9951? 
•99530 
•99543 

11  .9291^ 
.94026 
.95167 
•96338 
•97541 

20 

21 

22 
23 
24 

25 
26 

27 
28 

29 

9.98783 

.98796 
.98809 
.98822 
.98835 

i  i  .  54643 
.55116 

•55593 
.  56076 

•56563 

9-99555 
•99568 
.99581 
•99594 
•99605 

1  I  .  98777 
I  2  .  0004£ 

.01358 
.02707 
•  04098 

11 

27 
28 

29 

30 

3i 
32 
33 
34 

9.98848 
.98861 

.98874 
.98887 
.  98900 

11.57056 

•57554 
.58058 

•58567 
.  59082 

9.99619 
.99632 
.99645 
.9965? 
.  99676 

12.05535 

.07020 

.08557 

.  10149 

.11801 

ao 

31 
32 

33 
34 

P 

37 
38 
39 

9.98913 
.98925 
.98938 
.98951 
.  98964 

i  i  .  59602 
.60129 
.60662 
.61202 
.61747 

9-99683 
•99695 
•99708 
•99721 
•99734 

12.13517 
.15302 

.17163 

.19105 

.21139 

35 
36 
37 
38 
39 

40 

4i 
42 

43 

44 

9-9897? 
.98996 
.99003 
.99016 
.99029 

ii  .62300 
.62859 

•63425 
.63998 

.64579 

9-99746 
•99759 
.99772 

•99784 
.  -9979? 

12.23271 
.25511 
.27872 
.3036? 
.33013 

40 

4i 
42 
43 
44 

45 
46 
47 
48 

49 

9.99042 
.99055 
.99068 
.99081 
.99093 

ii  .65167 
.65762 
.66365 
.66978 

•67598 

9.99810 
•99823 
.99835 
.99848 
.99861 

12.35828 

.3883? 

.42068 

•4555? 
•49349 

45 
46 
47 
48 

49 

50 

5i 
52 
53 
54 

9.99105 
.99II§ 
.99132 

•99H5 
.99158 

ii  .68227 
.68865 
.69511 
.70168 
.70834 

9.99873 
.99886 

.99899 
.99911 

•99924 

12.53501 
.  58089 
.63217 
.69029 
.75736 

50 

5i 

52 
53 
54 

55 
56 
57 
58 
59 

9.99171 
.99184 
.99197 
.99209 
.99222 

11.71509 
.72196 
.72892 
.73600 
.74319 

9-99937 
•99949 
.  99962 

•99974 
•9998? 

12.83667 
•93371 
13.0587? 
•23499 
•53615 

7931 
9704 
12505 
17621 
30116 

55 
56 
57 
58 
59 

60 

.9.99235 

11.7505° 

10.00000 

Infinity 

60 

Log.  V«r». 

Z) 

Log.  Kxsee. 

7> 

LOST.  Vers. 

7> 

jog.  Exsec. 

j> 

' 

P.  P. 

438 


TABLE  IX.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 


o       / 

Sin. 

d. 

Tan. 

d. 

Cot. 

d. 

Cos. 

d. 

] 

».  P. 

0  0 

O.OOOO 

O.OOOO 

00 

I.OOOO 

0  90 

10 
20 

30 
40 

50 

0.0029 
0.0058 
0.008? 
O.CIlg 

0.0145 

29 
29 
29 
29 

0.0029 
0.0058 
0.008? 
O.OIlg 

0.0145 

29 
29 
29 
29 
29 

343-773 
171.885 
114.583 
85.9398 
68.7501 

1.  0000 
I.GOOO 
0.9999 

0.9999 
0.9999 

o 
6 
o 
6 

50 

40 

30 

20 
10 

1  0 

0.0174 

^ 

0.0174 

29 

57.2899 

0.9998 

0  89 

30 

2Q 

29 

10 
20 
30 
40 
50 

0.0203 
0.0232 
0.0262 
0.0291 
0.0320 

29 
29 
29 
29 
29 

0.0203 

0.0233 

0.0262 
0.0291 
0.0320 

29 
29 
29 
29 
29 

49.1039 
42.9641 
38.1884 

34.367? 
31.2416 

6.1398 

4-7756 
3-8217 
3.1261 
f.f.-~ 

0.9998 
0.999? 

0-9996 
0.9996 
0.9995 

6 

i 
6 

5o 
40 
30 

20 
10 

i    3.0 

2     O.O 

3    9-o 
4  12.0 
515-0 
6  18.0 

2.9 
5-9 
8-8 
ii.  8 
14-7 
17.7 

2.9 
5-8 
8.7 

n.6 
'4-5 
'7-4 

2  0 

0.0349 

^y 

0.0349 

^y 

28.6362 

2  .  0053 

0.9994 

0  88 

7  21.0 

20-6 

23  6 

zo.3 

10 
20 
30 
40 
50 

0.0378 

0.0407 
0.0436 
0.0465 

0.0494 

^y 
29 
29 
29 
29 

0.0378 
0.040? 
0.0436 
0.0466 
0.0495 

^y 
29 
29 
29 
29 

26.4316 
24.541? 
22.903? 
21.4704 
20.2055 

i.889§ 
1.6380 
1-4333 
1-2643 

0-9993 
0.9991 
0.9996 

i 

i 
i 

i 

5o 
40 
30 

20 
10 

927.0 

26.5 

26.1 

3  0 

0.0523 

ay 

0.0524 

29 

19.0811 

1.1244 

0.9985 

i 

0  87 

10 
20 

30 
40 

5° 

0.0552 
0.0581 

0.0616 
0.0639 
0.0663 

*y 
29 
29 
29 
29 

0-0553 
0.0582 
0.0611 
0.0641 
0.0670 

*y 
29 
29 
29 
29 

18.0750 
17.1693 

16.3498 
15.6048 
14.9244 

9056 
8i95 
7455 
6804 

0.9984 

0-9983 
0.9981 

0-9979 
0-997? 

i 

i 

2 
2 

5o 
40 
30 

20 
10 

2§ 

I      2.8 

;« 

4  11.4 

5   * 

0.5  o. 

I.O  O. 

i-5  i. 

2.0  I. 

i   4 

40.4 

o  0.8 

§1-2 

81.6 

4  0 

0.069? 

••^y 

0.0699 

*9 

14.3006 

6237 

0-9975 

2 

0  86 

5  14-2 

2-52. 

22.0 

10 

20 
30 
40 

50 

0.0725 
0.0755 
0.0784 
0.0813 
0.0842 

^y 
29 
29 
29 
29 

0.0723 
0.0758 
0.0787 
o.o8ig 
0.0845 

*y 
29 
29 
29 
29 

13.726? 
13.1969 
12.7062 
12.2505 
11.826! 

5739 
5298 
49°7 
4557 
4243 

0.9973 
0-9971 
0.9969 
0.9967 
0.9964 

2 
2 
2 
2 
2 

5o 
40 
30 

20 
10 

o  17.1 

7  19-9 
822.8 
9  25-6 

3-0  2. 

3-53- 
4-03- 
4-54- 

7  2-4 

I  2.8 

63-2 
6  3.6 

5  0 

0.0871 

29 

0.0875 

ay 

11.4306 

3961 
•3706 

0.9962 

0  85 

10 

20 

30 

40 

50 

0.0900 
0.0929 
0.0958 
0.0987 
0.1015 

29 
29 
29 
29 

0.0904 
0.0933 
0.0963 
0.0992 

0.102! 

29 
29 
29 
29 

11.0594 
10.7119 
10.3854 
10.0786 
9.7881 

3475 
3265 
3073 
2899 

0-9959 
0-9956 
0.9954 
0.9951 
0.9948 

3 

2 

3 
3 

50 
40 

30 

20 
10 

io.§d 

2  0.7  0 

?,A 

.60.5 

2 

0.2 

o-4 

e  o 

0.1045 

O.I05I 

9.5M3 

0.9945 

3 

0  84 

3  i.  60 

.90.7 

0.6 

10 
20 

30 

40 

50 

0.1074 
0.1103 
0.1132 
o.  1161 
0.1190 

28 
29 
29 
29 
29 
_; 

O.IO86 
0.  1  1  10 

0.1139 

0.1169 
o.i  198 

29 
29 
29 
29 
29 

9-2553 
9.0098 
8.7769 
8-5555 
8-3449 

2454 
2329 
2213 
2106 

0.9942 
0.9939 

0.9935 
0.9932 
0.9929 

3 
3 
3 
3 
3 

50 
40 
30 

20 
10 

•41-41 
51-71 

6  2.1   I 

7  2.4  a 

8  2.8  2 

93.12 

.2  I.O 

iSiis 

.11.7 
-42.0 
.72.2 

0.8 

I.O 
1.2 

i-4 
1.6 
1.8 

7  0 

O.I2I8 

28 

0.1228 

29 

8.1443 

0.9925 

3 

0  83 

10 
20 
30 
40 
50 

0.1247 

O.I27S 

0.1305 

0.1334 
0.1363 

29 
29 

2§ 
29 

0.1257 
0.1287 
o.i3i§ 
o.  1  346 
0.1376 

29 
29 
29 
3° 

7-9530 
7.7703 
7-595? 
7-4287 
7.268? 

i82g 
1746 
1670 
1599 

0.9922 
0.9918 
0.9914 
0.9916 
0.9905 

3 

4 
3 
4 
4 

50 
40 

30 

20 
10 

| 

8  0 

O.I391 

28 

o.  1405 

_3 

7-  "53 

J534 

0.9902 

4 

0  82 

io.5 

O.I  0 

.5 

10 
20 
30 
40 
50 

0.1420 
0.1449 
0.1478 
0.1507 
o.i53§ 

29 

2§ 
29 

28 

0.1435 

o.  1465 

0.1494 
0.1524 
0.1554 

3° 
29 
30 

29 

6.9682 
6.8269 
6.6911 
6.5605 
6-4348 

1413 
1358 
1306 
1257 

0.9898 
0.9894 
0.9890 
0.9886 
0.988! 

4 
4 
4 
4 
4 

5o 

40 

30 

20 
10 

2  0.- 

3  OH 
4o.e 
50.7 
60.5 

7  i.c 

0.2  0 
'0.30 

0.40 

0.5  o 
0.6  o 

0.70 

.1 

.1 

.2 

.2 

•3 
•3 

9  0 

0.1564 

29 

0.1584 

3° 

6.313? 

0.9877 

4 

0  81 

8  1.5 
0  I  : 

0.80 

•\ 

10 

20 
30 
40 

50 

o-i593 
0.1622 
0.1656 
0.1679 
0.1708 

2§ 
29 

2§ 
2§ 
29 

0.1613 
o.  1643 
0.1673 
0.1703 

0.1733 

29 
3<> 
3° 
3° 

3° 

6.1976 
6.0844 

5-975? 
5.8708 

5-7693 

1107 
1126 
1087 
1049 
1014 

0.9872 
0.986? 
0.9863 
0.9858 
0.9853 

4 
5 
4 
5 
5 

5o 
40 
30 

20 
10 

10  0 

0.1736 

10 

28 

0.1763 

3° 

5-6713 

goo 

0.9848 

5 

0  80 

Cos. 

d. 

Cot. 

d. 

Tan. 

d. 

Sin. 

d. 

/         0 

] 

P.  P. 

8O°-9O 


439 


TABLE  IX.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

1O°-2O° 


0       / 

Sin. 

d. 

Tan. 

d. 

Cot.            d. 

Cos. 

d. 

p.p. 

1  10  0 

10 
20 

30 
40 

50 

11  0 

10 

20 
30 
40 

50 

12  0 

10 

20 
30 
40 
50 

13  0 

10 
20 
30 
40 
50 

14  0 

10 
20 
30 
40 
50 

15  0 

0.1736 

2§ 

2§ 
29 
28 
28 
2§ 
2§ 
28 
2§ 

28 

2§ 
2§ 
2§ 
2§ 
2§ 
2§ 
28 

2§ 
28 
28 

2§ 
23 

28 
28 

28 

28 

2§ 

28 
28 
28 

0.1763 

30 
3° 
3° 
3° 
3° 
36 
30 
3° 
3° 
35 
30 
30 
.30 
33 
30 
30 
30 
30 

31 
30 
3i 
30 
3i 
31 
35 
3i 
31 
31 
3? 
31 
3i 
3i 
3i 
3* 
3? 
3? 
3i 
3? 
3i 
33 
3i 
32 
35 
32 
32 
32 
32 
32 
32 
32 

32 
32 

32 
32 
32 
32 
35 

33 
33 
32 

5-67I3 

949 

'      919 
890 
862 

«36 
811 

787 
764 
742 
721 
701 
682 
664 
646 
629 
613 
597 
582 
568 
553 
546 
527 
5i5 
5°2 
491 
480 
469 

458 
449 
439 
429 
420 
411 
4°3 
394 
387 
379 
37i 
364 
357 
350 
343 
337 
33i 
324 
3i9 
3i3 
307 
302 

296 
291 
28g 
281 

277 
272 
267 
263 
259 
254 

0.9848 

5 
5 
5 
5 
5 
5 
6 

5 
6 
5 
6 
6 
6 

6 
6 

6 
6 
6 
6 
6 
7 
6 
7 
7 
7 
7 
7 
7 
7 
7 
8 
7 
7 
8 
8 

a8 

I 
1 

J 

5 

8 

J 

8 

9 
9 
9 

' 

9 

9 

! 
5 

? 
? 

9 

x: 
9 

10 

080 

50 
40 
30 

20 
10 

079 

5o 
40 
30 

20 
10 

078 
50 
40 
30 

20 
10 

0  77 
5o 
40 
30 

20 
10 

076 

5o 
40 
3o 

20 
IO 

0  75 

5o 
40 
30 

20 
10 

0  74 

5° 

40 

30 

20 
10 

0  73 

5o 
40 
30 
20 

10 

072 

50 
40 

30 

20 
10 

0  71 

5o 
40 
30 
20 

10 

0  70 

I 

2 

3 

4 
5 
6 

7 
8 

9 

i 

2 

3 
4 

7 

8 

9 

i 

2 

3 
4 
1 

I 
9 

< 

I  C 

2 

3  : 

4: 

6  4 

5< 

9< 

33 

1:1 

9-9 

13-2 
16.5 
19.8 

23.! 
26.4 
29.7 

36 

3-5 
6.1 
9.1 

12.2 

15.2 

18.3 

21.  § 
24.4 
27.4 

28 

2-8 

w 

11.4 
14.2 
17.1 

19.9 

22.8 
25-6 

10 

I  1.0 
2  2.0 

33-o 

^  4.0 

55° 
5  6.0 

7  7-o 
3  8.0 
jg.o 

?      ' 

>-7o 

•5* 

'.2  2 

32 

3-2 

6.4 

9.6 

12.8 

16.0 
19.2 

22.4 

25.6 
28.8 

30 

3.0 
6.0 
9.0 

12.  0 
15.0 

18.0 

21.  0 

24.0 
27.0 

28 

2.8 

5.6 

8.4 

II.  2 
14.0 

16.8 

19.6 
22.4 
25.2 

9 

0.9  c 

1.8   3 

2.7: 

3.6: 

4-5  1 
5-4  4 

6-3; 
7.26 
8.17 

7   6 

7Q.( 
4  i.: 
i  i.* 

31 

3.1 

6.2 

9-3 

12.4 
i5.5 
18.6 

21.7 
24.8 
27.9 

29 

2-9 

5-8 
8.7 

n.  6 
M-S 
17.4 

20.3 

3::    | 

27 

2.7 

5-4 
8.1 

10.8 
13-5 
16.2 

18.9 

21.6 

24.3 
8 

>.'8 
.6 
•4 

•2 
.0 

.8 

.6 
•4 

.2 

3.S,        ! 

'.  1.0 

*  i.  5 

0.1765 
0.1793 
0.1822 
0.1851 
0.1879 

0.1793 
0.1823 
0.1853 
0.1883 
0.1913 

5.5764 
5.4845 

5-3955 
5-3093 
5-2256 

0.9843 
0.9838 
0.9832 
0.9827 
0.9822 

O.IpoS 

0.1944 

5-I44S 

0.98lg 

0.1936 
0.1965 
0.1993 

0.2022 
0.2050 

0.1974 
0.2004 
o.  203^ 
0.2065 
0.2095 

5.0653 
4.9894 
4.915! 
4-843P 
4-7728 

0.9816 
0.9805 
0.9799 
0.9793 
0.978? 

0.2079 

0.2125 

47046 

0.9781 

0.210? 
0.2136 
O.2I64 
0.2193 
0.2221 

0.2156 
0.2185 
0.2217 
0.224? 
0.2278 

4.6382 

4-5736 
4.5107 

4-4494 
4-3897 

0.9775 
0.9769 
0.9763 
0.9756 
0.9750 

O.2249 

0.2303 

43315 

09743 

0.2278 
0.2306 

0.2334 
0.2362 
0.2391 

0.2339 
0.2370 
0.2401 
0.2431 
o.  2462 

4-2747 
4.2193 
4.1653 
4.1125 
4.0610 

0.9737 
0.9736 
0.972§ 
0.9717 

O.Q7IO 

0.2419 

0.2493 

4.0108 

0.9703 

0.244? 
0.2475 
0.2504 
0.2532 
0.2560 

0.2524 
0.2555 
0.2586 

0.2617 

0.2643 

3-96I6 
3-9136 
3.8667 
3.8203 
3-7759 

0.9696 

0.9688 

0.9681 
0.9674 

0.9665 

0.2588 

0.2679 

3.7326 

0.9659 

10 
20 
30 
40 
50 

16  0 

10 
20 

30 
40 

50 

17  0 

10 
20 
30 
40 
50 

18  0 

10 
20 
30 
40 
50 

19  0 

10 
20 
30 
40 
50 

120  0 

0.2615 
0.2644 
0.2672 
0.2706 
0.2723 

28 
28 
28 
28 
28 
28 
28 

27 

28 
28 

0.2710 

0.2742 
0.2773 

0.2804 

0.2836 

3.6891 
3.6476 
3.6059 

3.5655 
3-5261 

0.9651 
0.9644 
0.9635 
0.9628 

0.9626 

0.2756 

0.286? 

3.4874 

0.9612 

0.2784 
0.2812 
0.2840 
0.2868 
0.2896 

0.2899 
0.2936 
0.2962 

0.2994 
0.3025 

3-4495 
3-4123 
3-3759 
3-3402 
3-3052 

o.  9604 

0.9595 
0.9588 

0.9580 

0.9571 

02923 

^7 
28 
28 
27 
28 
27 
27 

28 

0.305? 

32708 

0.9563 

0.2951 
0.2979 
0.3007 

0.3035 
0.3062 

0.3089 
0.3121 

0.3153 
0.3185 
0.3217 

3-2371 
3.2046 
3.1716 

3-1397 
3.1084 

0-9554 
0.9546 

0-9537 
0-9528 
0.9519 

0.3090 

0.3249 

3.0777 

0.9516 

0.3118 
o.3i4§ 
0.3173 
0.3206 
o  3228 

27 

27 
27 
27 
27 
27 
27 
27 

27 
27 
27 

0.3281 

0.3313 
0.3346 

0-3378 
0.3411 

3-0475 
3-OI78 
2.9887 
2.9606 
2.9319 

0.9501 
0.9492 
0.9483 
0.9474 
0.9464 

•73-5  3-< 
•5  4-2  3-< 

.2  4.9  4.5 
).o  5.6  4.$ 
>-76.35-4 

>  2.5 

)3.0 

3-5 
4.0 

4-5 

0.3255 

0-3443 

2.9042 

0-9455 

03283 
0.3316 
0-3338 
0.3365 
0-3393 

0.3476 

o-35°8 
0.3541 

0.3574 
0.3607 

2.8770 
2.8502 
2.8239 
2.7980 

2.7725 

0.9445 
0.9436 

0.9426 
0.9415 
0.9407 

0.3420 

0.3639 

2-7475       '" 

0.9397 

Cos. 

d. 

Cot. 

d. 

Ta«.           d. 

Sin.          d. 

/     o 

P.P. 

7O°  8O 


440 


TABLE    IX.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

2O°-3O° 


o     / 

Sin. 

d. 

Tan.         d. 

Cot.           d. 

Cos. 

d. 

p.p. 

20  0 

10 
20 
30 
40 

5° 
21  0 

10 
20 
30 
40 
50 

22  0 

10 
20 
30 
40 

50 

23  0 

10 
20 
30 
40 
50 

24  0 

10 
20 
30 
40 
50 

25  0 

03420 

27 
27 
27 

27 

27 

27 

27 
27 

27 

27 
27 
27 
27 
27 
27 

26 
27 
27 

26 

27 

26 
26 

27 

26 
26 
26 
26 
26 
26 

26 

26 
26 

26 

26 

26 
26 

26 

26 
26 
26 
26 
26 
26 
25 
26 
26 
25 
25 
26 
25 
25 

25 

25 
25 

2§ 

25 
25 

25 
25 
25 

0.3639 

33 
33^ 
33 
33 
33 
33 
33 
33 
33 
33 
34 
34 
33 
34 
34 
34 
34 
34 
34 
34 
34 
35 
34 
35 
34 
35 
35 
35 
35 
35 
35 
35 
36 
35 
36 
36 
36 
36 
36 
36 
36 
37 
36 
37 
36 
37 
37 
37 
37 
37 
37 
38 
37 
38 
38 
38 
38 
38 
39 
38 

2-7475 

247 
242 
239 
235 
232 
228 
225 

221 

218 

215 
212 
20§ 
206 
203 
200 
197 
194 
I92 
I89 
I87 
I84 
182 
I79 
177 

J7S 
172 
170 
1  68 
166 
164 
162 
159 
158 
156 
'54 
'52- 
150 
J48 
H7 
MS 
'43 
142 
140 
'39 
137 
136 
134 
132 

*3i 
130 

I2§ 

127 

125 

124 

123 

122 
120 
II§ 

118 

117 

0-9397 

10 
10 
10 
10 
IO 
10 
10 
IO 

II 

16 
xi 

10 

II 
II 
II 
II 
II 
II 
II 
II 
II 

II 

12 
II 
12 
12 
12 
12 
12 
12 
12 
12 
12 
12 

X3 
12 

*3 

12 

13 
13 
13 

J3 
13 
i§ 

*§ 
!§• 

J3 
i§ 
i§ 
14 
14 
i§ 
»4 
14 
»4 
14 
H 
'4 
14 
»4 

0   70 

5° 
40 

30 

20 

10 

0  69 

50 
40 
30 

20 
10 

0  68 

50 

40 
30 
20 

IO 

0  67 

50 

40 

30 

20 
10 

0  66 

50 
40 
30 
20 

10 

0  65 

5o 
40 
30 

20 
10 

0  64 

50 
40 
30 

20 
10 

0  63 

50 
40 
30 

20 
10 

0  62 

50 
40 
30 

20 
10 

0  61 

50 
40 
30 

20 
10 

0  60 

I 

2 

3 

4 

i 

I 

9 

i 

2 

3 

4 

i 

9 

i 

2 

3 

4 
5 
6 

7 

i 

2 

3 
4 

i 

7 
8 

9 

39 

3-9 
7-8 
11.7 

15-6 
19-5 
23-4 

27-3 
31-2 
35-i 

35 

3-5 
7-i 
10.6 

14.2 
17.7 

"'3. 
3: 

31.9 

1, 

1:1 

II.  0 

J3-7 
16.5 

19.2 

22.0 
24.7 

14 

1.4 

2'§ 

4-3 

5-8 
7.2 
8-7 

lo.i 
n.  6 
13-0 

I 

38 

« 

ii.4 

15.2 
19.0 

22.8 

26.6 
30-4 

34-2 

35 

3-5 
7-o 
10.5 

14.0 
17-5 

21.0 

24-5 
128.0 
31-5 

27 

2-7 

5-4 

10.8 
'3-5 
16.2 

18.9 

21.6 

24-3 

14 

'•4 

2.8 

4-2 
5-6 

t: 

9-8 

II.  2 
12.6 

I     I 

37 

3-7 
7-4 

IO.I 

14.8 
18.5 

22.2 

25-9 
29.6 

33-3 

34 

11 

10.2 

13-6 
17.0 
20-4 

23.8 
27.2 
30.6 

26 

2.6 
5-2 

7.8 

10.4 
13.0 
15.6 

18.2 

20.8 

23.4 

13 

2.1 

3-9 

£ 

7-8 

9.1 

10.4 
11.7 

I     I 

36 

3-6 

7-2 

10.8 

14.4 
18.0 

21.6 

25.2 

28.8 
32-4 

33 

1:1 

9.9 

13.2 
16.5 
i9.8 

23.1 
26.4 
29.7 

25 

2.5 

5-0 
7-5 

10*.  0 

I2-5 
i5-o 

17-5 

20.0 

22-5 

12 

1.2 
2-4 

3.6 

4-8 
6.0- 
7.2 

8-4 
9-6 

10.8 

0 

0 

o 

0 

o 

0 
0 

0 

o 

0 

0.3447 
0.3475 
0.3502 
0.3529 

0-3558 

0.3672 
0.3705 
0.3739 
0.3772 
0.3805 

2.7228 
2.6985 
2.6746 
2.6511 
2.6279 

0.9387 
0.9377 
0.9366 
0-9356 
0.9346 

03583 

0.3838 

2.6051 

0.9336 

0.3611 
o.  3638 
0.3665 
0.3692 
0.3719 

0.3872 
0.3905 

0.3939 
0.3972 

0.4006 

2.5826 

2.  5604 

2.5386 
2.5I7I 

2.4959 

0.9325 

0.9315 
0.9304 
0.9293 
0.9282 

0.3746 

0.4040 

24751 

0.9272 

0-3773 
0.3800 
0.3827 

0.3853 
0.3886 

0.4074 
0.4IO8 
0.4142 

0.4I7S 
0.4210 

2.4545 
2.4342 
2.4142 

2-3945 
2.3756 

0.9261 
0.9250 
0.9239 
0.9227 
0.9215 

0.390? 

0.4244 

2-3558 

0.9205 

0-3934 
0.3961 
0.398^ 
0.4014 
0.4041 

0.4279 

0.4313 
0.4348 

0.4383 

0.441? 

2-3369 
2.3182 
2.2998 
2.2815 
2.263? 

0.9193 
0.9182 
0.9176 
0.9159 
0.9147 

0.406? 

0.4452 

2.2466 

0.9135 

0.4094 
0.4126 
0.4147 

0.4173 
0.4200 

0.4487 
0.4522 

0.455? 
0.4592 

0.462? 

2.2285 
2.2113 

2.1943 
2.1775 
2.1609 

0.9123 
0.911! 
0.9099 
0.9087 
0.9075 

0.4226 

0.4663 

2.1445 

0.9063 

10 
20 
30 
40 
50 

26  0 

10 
20 
30 
40 
50 

27  0 

10 
20 
30 

40 

50 

28  0 

10 
20 
30 
40 
50 

29  0 

10 
20 
30 

40 

50 

30  0 

0.4252 
0.4279 
0.4305 
o.433i 
0.4357 

0.4693 

0-4734 
0.4770 
0.4805 
0.4841 

2.1283 
2.1123 

2!o6^5 

0.9056 

0.9038 
0.9026 
0.9013 
0.9006 

o.438§ 

0.487? 

2.0503 

0.8988 

0.4410 
0.4436 
0.4462 
0.4488 
0.4514 

0.4913 
0.4949 
0.4986 
o.  5022 
0.5053 

2.0352 

2.0204 

2.0057 
1.9911 
1.9768 

0.8975 
0.8962 
0.8949 
0.8935 
0.8923 

0.4540 

0509S 

1.9626 

0.8910 

0.4566 
0.4591 
0.4617 

0.4643 
0.4669 

0.5132 
0.5169 
0.5205 
0.5242 
0.5280 

1.9486 
1.9347 

1.9210 

1.9074 

1.8940 

0.8897 
0.8883 
0.8870 
0.8855 
0.8843 

0.4694 

o.53i7 

1.880? 

0.8829 

0.4720 
0.4746 
0.477? 
0.4797 
0.4822 

0-5354 
0.5392 
o.  5429 
o.546? 
0.5505 

1.8676 
1.8546 

1.841? 

1.8296 
1.8165 

0.8816 
0.8802 
0.8788 
0.8774 
0.8766 

2     2.3     2.2     2 

3    3-4    3-3    3 

5    5-7    5-5    5 
6    6.9    6.6    6 

7    8.6    7.7    7 
8    9.2    8.8    8 
9  10.3    9.9    9 

0.4848 

0-5543 

1.8046 

0.8746 

0-4873 
0.4899 
0.4924 
0.4949 
0.4975 
0.5000 

0.5581 
0.5619 

°-*6A 
0.5696 

0.5735 

1.7917 

1.7795 
1.7675 
1.7555 
1-743? 

0.8732 
0.8718 
0.8703 
0.8689 
0.8675 

0.5773 

1.7326 

o  8666 

Cos.         d. 

Cot. 

d. 

Tan. 

d. 

Sin. 

d. 

/       0 

P.P. 

6O  -7OC 


441 


TABLE  IX.— NATURAL  SINES,  COSINES,   TANGENTS,  AND  COTANGENTS. 

30°-4O° 


o    / 

Sin. 

d. 

Tan. 

d. 

Cot. 

d. 

Cos. 

d. 

p.  p. 

30  0 

IO 

20 
30 
40 
50 

:31  0 

10 

20 
30 
40 

50 

32  0 

10 
20 
30 

40 

50 

33  0 

10 
20 
30 
40 
50 

0.5000 

25 
25 
25 
25 
25 
25 
25 
24 
25 
25 
24 
24 
25 
24 
24 
24 

24 

24 

24 

24 
24 
24 

24 

24 
24 
24 
24 
24 
24 
24 
23 
24 

23 

23 

24 

0-5773 

39 
39 
39 
39 
39 
39 
40 
39 
40 
40 
46 
40 
40 
4i 
40 
41 
48 

4? 

42 

42 
42 
42 

42 
43 

43 
43 

1.7320 

116 
114 

"3 

112 
III 
110 
I0§ 

108 
107 
106 

104 
103 

102 
IOI 
100 

99 
98 
97 
96 
96 

95 
94 
93 
92 
92 

90 
89 

89 
88 
87 
86 
86 
85 
84 
84 
83 
83 

si 

81 
85 
80 
79 
78 
78 
77 
77 

76 
76 

75 
74 
74 
73 
73 
73 
72 

71 
76 

0.8666 

15 

14 
15 
15 
15 

15 
15 
15 

15 
15 

16 
15 
16 

i§ 
16 
16 
16 

*6 
16 

16 
'6 
16 

J7 

17 
17 

17 
17 
17 
17 
17 
if 
17 
17 
17 
18 

17 
18 

18 
18 
18 
18 
18 

'§ 
18 

'§ 

0  60 

50 
40 
30 
20 

IO 

059 

50 

40 

30 

20 
10 

058 

40 
30 

20 
IO 

057 

40 
30 

20 
10 

056 

40 
30 

20 
10 

055 

40 
30 

20 
10 

054 

50 
40 
30 

20 
10 

053 

40 
30 

20 
10 

052 

50 
40 
30 

20 
10 

051 

40 

30 
20 

IO 

050 

4 

I    4 

2    9 

524 
625 

734 
83C 
944 

A 

i    4 
2    9 
312 

41? 

5  22 
627 

940 

i 

2 

3 
4 

i 

9 

X 
2 

3 
4 

1 

9 

I 

2 

3 
4 

7 
8 

9 

I 

X     X 

2     3 

3   5 

\l 

6  10 

712 
814 
9H5 

5 

.8 
•7 

•  7 

i 

.6 
•5 

3 

-5 
.  i 

•6 

.2 

•7 
•3 

1 

4 

4 
8 

12 

16 

2C 
24 

29 

33 
37 

2 

2 

5 
7 

10 
12 
15 

20 
22 

2 

2 

4 
6 

9 
ii 

20 

7 

•7 

.0 

4 

4 
9 
14 

19 
24 
25 

34 
39 
44 

4 

A 
9 

ij 

18 

22 
27 

4° 

I 

.1 

•3 
•4 

.6 
-7 
•9 

.6 

.  i 
•6 

.2 

•3 
•8 

1 

2 

.2 

•5 
•7 

.0 

.2 

•5 

•7 
.0 

.2 
I 

3 
5 

6 

9 

•9 
.8 

•7 

.6 
•5 
•4 

-3 

.2 
.1 

5 

•5 

.0 

•5 

.0 

•5 

.0 

•5 

.0 

•5 

4 

4 
8 

12 

16 

20 
24 

28 

H 

2 

2 

5 
7 

10 
12 
15 

17 

20 
22 

2 

2 

3 

S 
13 
15 
19 

7 

•4 
.  i 

.8 

4 

4 
9 
M 

19 

3 

51 

43 

4 

13 

22 
26 

3° 
35 
39 

I 

.2 

•3 

•4 
•  5 
.6 

•7 
.8 
•9 

5 

•5 

.0 

•5 

.0 

•5 

.0 

•  5 

.0 

•5 
2 

.2 

•4 
.6 

.8 

.0 
.2 

•4 
6 
.8 

I 

i 
3 
4 

6 

8 

.8 
.6 
•4 

.2 
.O 

.8 

.6 

•4 

.2 

4 

•4 
.8 

.2 

.6 
.0 

•4 

.8 

.2 

.6 

4 

! 

12 

id 

2C 

24 

2£ 

H 

2 

2 

4 
7 

S 

12 

15 
21 

T 

I 

3 
5 

7 

9 
ii 

12 

14 

16 

5 

6 

2 

8 
4 

4' 

4 
9 
M 

18 

11 

32 
37 
42 

4 

4 
8 

12 

17 
21 

25 

3° 

H 

0 

.0 
.0 
.0 

.0 

.0 

.0 

.0 
.0 
.0 

4 

1 

.2 

.6 

.0 

•4 

.8 

.2 

.6 

I 

•8 

•  4 

.2 

.1 

1 
1^ 

I 
6 

1  46 

•7    4-6 
.4    9-2 
.113-8 

.818.4 
•523.0 
.227.6 

.932-2 
.636.8 
•3J4X«4 

3  42 

•3    4-2 
.6    8.4 
.9  12.6 

.2  16.8 

•5  21.0 

.825.2 

.1  29.4 

•433-6 
•737-8 

3-9 
11.7 

15-6 
19-5 

23-4 

27-3 
31-2 
35-1 

23 

6'.g 

9-2 

"•5 
i.H.S 

16.1 
18.4 
20.7 

18 

x.a 
3.6 

5-4- 

7.  a 
9-0 
10.8 

12.6 

14.4 

16.2 

*5    i-4 

o    2.9 
5    4-3 

o   6.8 

0.5025 

o.  5056 

0.5075 
0.5106 
0.5125 

0.5812 
0.5851 
0.5896 

0.5929 
0.5969 

1.7204 
1.7090 
1.6975 
1.6864 
1.6753 

0.8645 
0.8631 
0.8615 
0.8601 

0.5156 

~  0.6048 
0.6088 
0.6128 
0.6168 
0.6203 

1.6643 

0.857* 

0.5175 
0.5200 
0.5225 
0.5250 

0.5274 

1.6533 
1.6425 

1.6318 
1.6212 
1.610? 

0.8555 
0.8541 
0.8525 
0.8511 
0.8496 

0.5299 

0.6243 

1.6003 

0.8486 

0.5324 

0-5348 
0.5373 
0-539? 
0.5422 

0.6289 
0.6330 
0.6376 
0.641! 
0.6453 

1.5900 
1.5798 
1.5697 

1-5596 
1-5497 

0.8465 
0.8449 
0.8434 
0.8418 
0.8402 

0.5446 

0.6494 

1-5398 

0.8386 

0.5471 
0.5495 
0.5519 

0.5543 
0.5568 

0.6535 
0.6577 
0.6619 
0.6661 
0.6703 

1.5301 
1.5204 

1.5108 
1.5013 
1.4919 

0.8371 
0-8355 
0.8339 
0.8323 
0.8305 

34  0 

0.5592 

0.6745 

1.4823 

0.8296 

10 
20 
30 
40 
50 

0.5616 
o.  5640 
o.  5664 

0.5688 
0.5712 

0.678? 
0.6830 
0.6873 
0.6915 
0.6959 

1-4733 
1.4641 
1.4550 
1.4460 
I.437Q 
1.4281 

0.8274 
0.825? 
0.8241 
0.8225 
o.  8208 

85  0 

0.5736 

0.7002 

0.8191 

IO 

20 
30 
40 
50 

;36  o 

IO 

.20 
30 

j     40 

37  G 

0.5759 
0.5783 
0.5807 
0.5836 
0.5854 

0.7045 
0.7089 

0.7133 
0.7177 
0.7221 

43 
44 
44 
44 
44 
44 
44 
45 
45 
43 
43 
4§ 
46 

46 
46 

46 
47 
47 
47 
47 
47 
48 
48 
48 
48 
49 
49 
49 
49 

I.4I93 
1.4106 
1.4019 
1-3933 
1-3848 
1-3764 

0.8175 
0.8158 
0.8141 
0.8124 
0.8107 

0.5878 

23 
23 
23 
23 

2§ 

23 

2§ 
23 
23 
23 
23 
23 
23 
23 
22 

23 
22 
22 

23 
22 
22 
22 
22 
22 

0.7265 

0.8090 

0.5901 
0.5925 
0.5948 
0.5971 
0.5995 

0.7310 
0.7353 
0.7399 

0.7490 

1.3680 
1.3597 
I.35I4 
1.3432 
I.335I 

0.8073 
0.8056 
0.8033 
0.8021 
0.8004 
0.7986 

0.6018 

0-7533 

1.3276 

IO 
20 

30 
40 
50 

38  0 

10 

20 
30 
40 
50 
39  0 

10 

20 
30 
40 
50 
40  0 

0.6041 
0.6064 
0.608? 
0.6116 
m  0.6133 
0.6156 

0.7581 
0.7627 
0.7673 
0.7719 
0.7766 

1.3196 
1.3111 
1.3032 
1.2954 
1.2875 

0.7969 
0.7951 

0-7933 
0.7916 
0.7898 

0.7813 

0.7860" 
0.7907 

0-7954 
0.8002 
0.8050 

1.2799 

0.7880 

0.6179 
©.6202 
0.6225 
0.6248 
0.6276 

1.2723 
1.2647 
1.2571 
1.2497 
1.2422 

0.7862 
0.7844 
0.7826 
0.7808 
0.7789 

0.6293 

0.8098 

1-2349 

0.777* 

0.6316 
0.6333 
0.6361 
0.6383 
o.  6405 

0.8146 
0.8194! 
0.8243 
0.8292 
0.8341 

1.2276 
1.2203 
1.2131 
1.2059 
1.1988 

0-7753 
0.7734 
0.7716 
0.769? 
0.7679 

•5 

.2 

:? 

xo.aj  9.6 

11.9:11.2 
13.6  12.8 
I5-3I4-4 

7-5    7-2 
9.0    8.7 

10.5  10.  i 

12.0  II.  6 

13-513-6 

0.6428 

0.8391 

1.1919 

0.7666 

Cos. 

d. 

Cot. 

d. 

Tan. 

d. 

Sin. 

d. 

/         0 

p.p.          ! 

5O°-6O 


442 


TABLE   IX.— NATURAL  SINES,  COSINES,  TANGENTS,  AND  COTANGENTS. 

40°-4:50 


'    o     / 

Sin. 

d. 

Tan. 

d. 

Cot.        d. 

Cos. 

d. 

P.P. 

400 

0.6428 

0.8391 

19 

1.191^    70 

0.7666 

IQ 

0 

50 

10 

20 
30 

4° 

0.6450 
0.6472 
0.6494 
0.6515 

22 
22 
22 

0.8446 
0.8496 
0.8541 
0.8591 

5° 

50 
50 

51 

'$71 

.1703 
.1640 

70 
69 

68 

0.764! 
0.7623 
0.7604 
0.7585 

18 
19 
19 

10 

50 
40 
30 

20 

i 

2 

3 

70 

14- 

21. 

22"       22      21       21 

D 

0 

4 
fi 

•I 
•1 

4-4 
6.6 

6'-! 

4-2 

6-3 

So 

0.6533 

7.2 

0.8642 

Si 

.157? 

68 

0.7566 

10 

28  o 

8  8 

8  6 

8   A 

410 

10 
20 

30 
40 

0.6566 

22 
22 
21 
22 
7T 

0.8693 

11 

52 

15 

•1503 

67 
67 
66 
66 

0-7547 

19 
19 

19 
19 

T9 

049 

50 
40 
30 

20 

I 

9 

35-o 
42.0 

49.0 
56.0 
63.0 

II 
13 

3 

20 

.2 

•: 
7 

.  0 
.2 

II.  0 

13.2 

15-4 
17.6 
19.8 

To.j 
12.9 

15-5 
17.2 
19-3 

10.5 

12.6 

0.6582 
0.6604 
0.6626 
0.6648 

0.8744 

0.8847 
0.8899 

•1436 
.1369 

•1303 
.1237 

0.7528 
0.7509 
0.7489 
0.7476 

So 

0.6669 

7.7, 

0.8951 

52 

.1171 

65 

0.7451 

10 

10 

420 

0.6691 

0.9004 

.1106 

0-743* 

048 

69      20       20      I§       19 

10 

20 

30 
40 
50 

0.6713 

0.6734 
0.6756 
0.677? 
0.6798 

21 
21 
21 
21 
71 

0.9057 
0.9110 
0.9163 
0.9217 

0.9271 

53 
53 

11 

54 
54 

.1041 
.0977 
.0913 
.0849 
.0786 

64 
64 
63 

63 

0.7412 
0.7392 
0-7373 
0.7353 
0-7333 

19 
19 

20 

19 
7O 

50 
40 
30 

20 
10 

2 

3 

4 

i 

6.9 
13-8 
20.7 

27.6 

34-5 
41.4 

2.6 

ti 

8.2 
10.2 
12.3 

2.0 
1° 

6.0 
8.0 

10.  0 
12.0 

3:? 
5-8 

7-8 
9-7 
11.7 

1.0 

3-8 
5-7 

7.6 

9-5 
11.4 

430 

10 
20 

30 
40 
50 

0.6820 

21 
21 
21 
21 
21 
2  1 

0.9325 

54 

55 

1 

.0723 

62 
62 
61 
61 
66 
60 

0.7313 

20 
20 
20 
20 
20 
2O 

0 

50 
40 
30 

20 
10 

47 

I 

9 

48-3 

68 

!4.§     14.0 
16.4     16.0 
18.4     18.0 

68    67 

«3-6 
15.6 

17-5 

66 

13-3 
15.2 
17.1 

18 

0.6841 
0.6862 
0.6883 
0.6904 
0.6925 

0.9379 

0-9434 
0.9489 
0.9545 
0.9601 

.0661 
.0599 
.0538 

•0476 
.0416 

0.7293 
0.7273 
0.7253 

0.7233 
0.7213 

440 

0.6948 

0.9657 

J 

10355 

0.7193 

046 

i 

2 

; 

7 

13.6 

0.7 
13-4 

13-2 

3-7 

10 

0.6967 

21 

0.9713 

56 

-2 

1.0295 

60 

0.7173 

20 

5° 

3 

20. 

5 

20.4 

20.1 

19.8 

5-5 

20 

30 
40 

0.6983 
0.7009 
0.7030 

21 
20 

21 

0.9770 
0.9827 
0.9884 

56 

i 

1.0235 

1.0176 
1.0117 

59 
59 
59 

0.7153 
0.7132 
0.7112 

2O 
20 
20 

40 
30 
20 

4 

i 

27. 

34- 
41- 

4 

I 

-7 
34 
4'- 

.2 

.0 

.1 

26.8 

33-5 
40.2 

26.4 

33-o 
39-6 

11 

ii.  i 

50 
450 

0.7050 

2O 
20 

0.9942 

J>y 
58 

1.0058 

58 
58 

0.7091 

2O 
20 

10 

0 

45 

i 

9 

47- 

11: 

S 

5 

47-6 
54-4 
61.2 

46.9 

I3/6 
60.3 

46.2 
52.8 
59-4 

12.9 
14-8 
16-6 

0.7071 

I.OOOO 

I.OOOO 

0.7071 

Cos. 

d. 

Cot.         d. 

Tan.     >  d. 

Sin.      |  d. 

f     o 

65  64 

64     63    62 

6i   66    59  59    58   58 

Si 

57  56 

56   55    54 

54 

53    53  52    52 

j 

6.5    6.Z 

6-4 

6  3 

6.2 

6.1     6.61 

5-9    5-9 

s-fi 

S-8 

5-7 

5-7 

5-6 

5. 

6     5-5 

5-4 

5-4 

5-3    5-3'    5-2 

5-2 

2 

3 

il'-l 

SI 

12.8 

19.2 

12.6 

18.9 

12.4 
18.6 

12-3    12.  I 

18.4  18.1 

11.9  ii.  8 
17.  §  17.7 

••I 

'7-5 

ii.  6 

£1 

11.4 
17.1 

"-2 
16.9 

16. 

2    II  . 

8  16. 

10.9 
16.3 

10.8 
16.2 

10.7  10.6  10.5 
16.6  15.9  15.7 

10.4 
15-6 

4 
S 

26.2 

12.  f 

25.8 

25.6 

32.0 

25.2 

24.8 
31.0 

24.6  24.2 
30.7  30.2 

23.8  23.6 
29.7  29.5 

*3-i 

23.2 

29.0 

3:? 

22.8 

28.5 

22.6 

28.2 

22. 

28. 

4    «2- 

D    27. 

21.8 

27.2 

21.6 

27.0 

21-4    21.2    21.0 

26.7  26.5  26.2 

20.8 

26.0 

6 

39-3 

38-7 

38.4 

37-8 

37-2 

36-9  36-3 

35-7  35-4 

J5-i 

34-8 

34-5 

34-2 

33-9 

33- 

6  33- 

32-7 

32.4 

32.1    31.8    31.5 

31.  2  1 

7 

4^8 

45-i 

44 

.8 

44-1 

43-4 

43.6  42.3 

4i-6  41-3 

»O.Q 

40.6 

40.2 

39-9 

39-5 

» 

2  38. 

381 

37-8 

37-4  37-1  36.7 

36-4 

fel 

51.6 

51.2 
57-6 

50-4 
56.7 

49-6 
55.8 

49.  2  48.  4 

55.3154.4 

47-6  47-2 
53-5  53-1 

|6.8 
52-6 

46.4 
52.2 

46.0 

45-6 

45-2 
5<>-8 

44- 
53- 

4M9-5 

43-6 
49-6 

3:S 

42.8  42.4  42.0 
48.1  47-7  47-2 

41.6 
46.8 

Table  for  passing  from  Sexagesimal  to  Circular 

Measure. 

0      Circular  Meas. 

, 

Circular  Meas. 

"     Circular  Meas. 

5* 

51   50 

50     49 

X 
2 

5- 

10. 

! 

10.2 

5.6 

10.  1 

5.0 

10.  0 

100 

1-74  532  9 

10 

o.oo  290 

5 

10     o.oo  004  8 

3 

»5- 

4 

15-3 

15.5 

15.0 

'4-8                 200 

3-49  065  3 

2O 

o.oo  581 

S 

20 

0.00009  7 

4 

20. 

fl 

20.4 

2O.  2 

20.0 

19.8                300 

5.23  598  8 

30 

o.oo  872 

3 

30 

0.00014  5 

5 

25- 

7 

25-5 

25.2 

25.0 

24.7 

40 

o.oi  163 

j 

40 

o.oo  019  4  i 

6 

30-9 

30.6 

30-3 

30.0 

40 

0.69813  i 

7 

36.6 

35-7 

35-3 

35-o 

34-6                   50 

0.87  266  4 

50 

o.oi  454  4 

50 

0.00  024  2    ! 

8 

41- 

2 

40.8 

40.4 

40.0 

39-6                   60 

1.04  710  f 

9     46. 

3 

45-9      45-4 

45-0      44-5 

6 

o.oo  174 

^ 

6 

o.oo  002  9  j 

70 

1.22  1730 

7 

o.oo  203  6 

7 

o.oo  003  4 

80 

1.396263 

8 

o.oo  232  7 

8 

o.oo  003  9 

00 

1-570796 

9 

o.oo  261  8 

9     o.oo  004  3 

45C-5O 


443 


TABLE    X.— NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS. 

O°-1O°  1O°-2O° 


0      / 

Yers. 

d. 

Exsec. 

d. 

6 

i 

2 

3 
3 

4 

f 

8 

8 

TO 
10 
II 
12 
13 
14 
H 

16 
16 

18 
19 

20 
21 
21 

22 
23 
24 
25 
26 
27 
2? 
2§ 

30 

31 
32 

33 
34 
34 
35 
36 
37 
38 
39 
40 

41 
42 

43 
44 
45 
46 

47 
48 

49 
50 

$ 

c    / 

Vers.   d. 

Exsec.   d. 

P. 

P. 

0  0 

10 
20 

30 
40 
5o 
1  0 

10 

20 

30 

40 

So 
2  0 

10 
20 

30 
40 
5o 
3  0 

10 
20 

30 
40 
So 

.OOOOO 

6 

i 

2 

4 

5 

6 

8 
8 

10 

16 
ii 

12 
13 
13 
15 
15 

16 

18 

19 
20 

.OOOOO 

10  0 

10 
20 
30 
40 
50 

11  0 

10 

20 
30 
40 
50 

12  0 

10 

20 
30 
40 
50 

13  0 

10 
20 

30 

40 

5° 
14  0 

10 
20 
30 

40 

5° 
15  0 

10 

20 
30 
40 
50 

16  0 

10 
20 
30 
40 
50 

17  0 

10 
20 
30 
40 
50 

18  0 

10 
20 
30 
40 
50 

19  0 

10 
20 
30 
40 
50 

20  0 

.01519 

51 

52 
52 

53 
54 
55 
55 
57 

II 
S 

61 

62 
62 
63 
64 
65 
66 

68 
69 
70 

76 

72 
72 
72 
74 
75 
75 
76 

78 

79 
80 

86 
8T 
82 
83 
84 
84 

85 
86 

8? 
8? 

89 
89 

90 

93 
93 
94 

95 
95 
97 

9? 
98 

.01542 

52 

53 
54 
55 
56 
57 
58 

59 
60 
61 
62 
62 

63 
65 
65 
66 
6? 
6§ 
69 
70 
7? 
72 
73 
74 

3 

77 
78 

79 
80 

8T 
82 

83 
84 
85 
86 
87 
88 
89 
96 

92 
93 
95 
95 
96 
98 
98 

100 
101 
IO2 
103 
104 
105 

log 

10? 

109 
109 

III 

112 

110 

i  ii 

2   22 

3  33 

4  44 

5  55 
6  66 

7  77 
8  88 
9  99 

30 

1   3 

2    6 

3   9 
4  12 
6  18 

7  21 
8  24 
9  27 

i 

2 

3 
4 

I 

9 

i 

2 

3 

4 
5 
6 

I 

9 

IOC 

10 

20 
30 

4° 

50 
60 

70 
80 
90 

20 

2 

4 
6 

8 

10 
12 

18 

7 

0.7C 

1.4  ] 

3.11 
2.8! 

3-5  : 
4-2  ; 

4.9, 

5-6. 
6-3. 

3 

0.3  « 

0.7  c 

1  .O  C 

1.4  1 
1.7  1 

2.1  I 

2.4  2 
2.8  2 
J.I2 

>9< 

c 

Ii 

2- 

3< 
4. 
g, 

6- 

E 

1C 

i 

2 

3 

4 

5 
6 

7 
8 

9 

§ 

).gc 

•3 
•9 

.6: 

•  S  ; 

•9 

h5  < 

,.2  , 
•§. 

3 

•3  < 

.6c 
•9  c 

.2 

•5 

,b 

.  i 

•4 
•72 

)fi 

> 

> 

I 
o 

2 

3 
4 
5 

6 

6 

.2 

.8 
2.4 

,.0 

j.6 
>-4 

2 

).2 

'•5 
'•7 

.0 
.2 

•5 

•  7 

.0 
.2 

0 

8 
16 
24 

32 
*° 
*8 

56 
72 

\ 

9  o 
9  i 

7060 

7[  6 

14  12 
21   l8 

28  24 

35  30 
42  36 

49  42 

56  48 
63  54 

9  8 

5040 

5   4 
10   8 

IS   12 

20   l6 
25   20 
30   24 

35  28 
40  32 
45  36 

8  f 

.80.7 

.6  1.5 

.00000 
.00001 

.00004 
.  00007 
.00016 

.00000 

.  ooooT 
.  00004 
.00007 

.00010 

.01570 
.Ol622 
.01674 
.01728 
.01782 

.01595 
.01643 
.01703 

•01758 
.01814 

.00015 

.00015 

.01839 

.01871 

.00020 

.00027 

.00034 

.  00042 
.00051 

.  00020 

.00027 

.00034 

.  00042 
.00051 

.01893 
.01950 
.  0200? 
.02066 
.02125 

.01929 
.01983 
.02043 
.02109 
.02171 

.00061 

.00061 

.02185 

.02234 

.00071 
.00083 

.00095 
.00103 

.00122 

.00071 
.00083 

.00095 

.00103 

.00122 

.02246 
.02308 
.02376 
.02434 
.02498 

.0229? 
.02362 
.02428 
.02494 
.02562 

•00137 

•00137 

.02563 

.02636 

.00152 
.00169 
.00186 
.00204 
.00223 

.00153 
.00169 
.00187 
.00205 
.00224 

.02629 
.02695 
.02763 
.02831 
.02906 

.02700 
.02770 
.02841 
.02914 
.02987 

83 
7  4 
7  5 

66 

5 

o«! 
i. 
*»i 

2.: 
B. 

3-: 
3-1 

<•: 

4'< 

2 
o.: 

O.I 

0.{ 

.c 

.2 

•A 
.t 
.8 

.6  3-43 
•5  4-2  4 
•45.1  4 

•35-95 
.26.8  6 
.1  7'67 

5  4 
'0.5  0.4 
i.  op.  5 

>  1-5  i.: 

'  2.0  I.E 

r  2.5  2.5 

J3-02.J 

53.53.5 

I.  4-o  3-^ 
>4.54-c 

I   I 

O.I  0.1 
0.3  0.2 
0.40.3 

0.6  0.4 
0.70.5 
0.9  o.( 

i.o  0.7 

T.2  O.g 
I.§O.S 

.23.0 
•03-7 
.84.5 

.65-2 
.46.0 
.26.7 

4 

o  4 
o  8 

I  2 

i  6 

2  O 
2  4 

2.8 

3-2 
3.6 

6 

0.6 

O.  I 
O.I 

0.2 
0.2 
0-3 

°'i 

[0.4 

4  0 

10 
2O 
30 
40 
50 

.00243 

.  00244 

.  02970 

.03061 

.  00264 
.00286 

.00331 

•00355 

21 

22 
23 
24 

.00265 
.00286 
.00309 
.00332 
.00357 

.03041 

.03113 
.03185 

•03258 
-03332 

.03136 
.03213 
.03290 
.03368 
•0344? 

5  0 

10 
20 

30 
40 

50 

6  0 

10 

20 

30 
40 

50 

7  0 
10 

20 
30 
40 
50 

8  0 

10 

20 
30 
40 
50 

9  0 

10 
20 
30 
40 
50 

10  0 

.00386 

26 
26 
2? 
28 

29 
30 
30 

P 

33 
33 
35 

.OO382 

•0340? 

•0352? 

.00406 
.00433 
.  00466 

.00518 

.00408 
•00435 
.00462 
.OO49I 
.00526 

.03483 
•03559 
.03637 

.03715 
•03794 

.03609 
.03691 
•03774 
.03858 
•03943 

.00548 

.00551 

.00582 
.00(514 
.00647 
.00681 
.00715 

.03874 

.04030 

•00578 
.006IO 
.00643 
.00676 

.  007  i  6 

•03954 
.  04036 
.04118 
.04201 
.04285 

.04117 
.04205 

.04295 
.04385 
.04476 

.00745 

.00751 

.04369 

.04569 

.00781 
.00818 
.00855 
.00894 
•00933 

36 
37 
38 
39 
40 

42 
42 
43 
44 

45 
46 

47 
4? 
48 
49 
50 

.0078? 
.00824: 
.00863 
.00902 
.00942 

.04455 
.04541 

.04716 
.04805 

.04662 

•0475? 
•04853 
.04949 
.0504? 

.00973 

.00983 

.04894 

.05146 

.01014 
.01056 
.01093 
.01142 
.01186 

.01024 
.OIO67 
.01110 

.01155 

.01200 

.  04984 
.05076 
.0516? 
.05266 
.05354 

.05246 
.05347 
•05449 
.05552 
.05656 

.01231 

.OI24g 

•05448 

.05762 

.01277 
.01324 
.01371 
.01420 
.01469 

.01293 
.01341 
.01396 
.01446 
.01491 

•05543 
.05639 
.05736 
•05833 
.05931 
.06036 

.05863 
.05976 
.06085 
.06194 
.06305 

.01519 

.OI545 

99 

.06418 

Exsec. 

d. 

0      / 

Vers.  i  d. 

Exsec. 

d. 

P. 

P. 

444 


TABLE    X.— NATURAL  VERSED  SINES  AND   EXTERNAL  SECANTS. 

2O°-3O°  3O°-4O° 


°  ' 

Vers. 

d. 

Kxsec. 

d. 

Vers.   d. 

Kxsec.   d. 

p.  P. 

20  0 

10 

20 

30 
40 

50 
21  0 

10 
20 

30 
40 

50 

22  0 

10 

20 

30 
40 

50 

23  0 

10 

20 

30 
40 

50 

24  0 

10 

20 

30 
40 
50 

|25  0 

10 
20 

30 
40 

50 

26  0 

10 
20 

30 
40 

50 

27  0 

10 
20 

30 
40 

50 

28  0 

10 
20 

30 
40 

50 

29  0 

10 
20 

30 
40 

50 

30  0 

.0603 

.0613 
.0623 
•0633 
.0643 
•0634 

.0674 
.0685 
.0696 
.0705 
.0717 

.0642 

II 
II 
II 
12 
II 
12 
12 
12 
12 
I  2 
I  2 
13 
12 
13 
13 
13 
13 
13 
13 
13 
14 
14 
H 
H 

H 
14 
H 
H 

!5 

15 
15 
15 

1$ 

\l 

I§ 

16 
16 
16 

16 
16 

17 
16 
17 
17 
17 
if 
17 
18 

18 
18 
18 
18 
18 
18 
19 
19 
19 
i§ 

30  0 

10 
-20 

30 
40 

50 

31  0 

10 
20 
30 
40 

50 

32  0 

10 

20 
30 
40 

50 

33  0 

10 

20 

30 
40 

50 

34  0 

10 
20 
30 

40 

50 

35  0 

10 
20 
30 
40 
50 

36  0 

10 
20 

30 
40 

50 

37  0 

10 
20 
30 
40 
50 

38  0 

10 
20 
30 
40 
50 

39  0 

10 

20 
30 
40 
50 
40  0 

•1339 
.1354 
.1369 

.1383  ! 
•1398 
•  1413 

!! 

H 
i5 
i5 
15 
i5 
15 
15 

15 
15 
i5 
15 

16 

15 
16 

16 
16 
16 

16 
16 

16 
16 
16 

16 
17 
16 
J6 
17 
17 
17 
17 
17 

17 
17 
If 
If 
17 

if 
18 

if 
if 
18 
18 
18 
18 
18 
18 

18 
18 

18 
18 
18 
18 
18 
i§ 

•1547 

19 
19 
20 
20 
20 
20 
20 
20 
21 
21 
21 
21 
21 
21 
22 
22 
22 
22 

23 
22 

23 
23 
23 
24 
24 
24 
24 
24 
24 

24 

% 

25 
2§ 
2$ 
26 

26 

27 
27 
2f 
2f 
2f 
28 

28 
28 
28 

29 

29 

29 

29 

30 
30 

3° 
30 

3i 

I 

i 

2 

3 

4 
5 
6 

7 
8 

9 

i 

31 

3.1 

6.2 

9'3 

12.4 
15-5 
18.6 

21.7 
24.8 
27.9 

27 

2.7 

30 

3-0 
6.0 
9.0 

12.0 

15.0 

18.0 

21  ,O 
24.0 
27.0 

26 

2.6 

29 

3 

8.7 

11.6 
'4-5 
'7-4 

20.3 
23.2 
26.1 

25 

2.5 

28 

2.8 

5-6 

8.4 

II  .2 

14.0 

16.8 

19.6 
22.4 
25.2 

24 

2.4 

10 

id 

10 
10 
10 
10 
10 

II 

10 

II 

10 

II 
II 
II 

II 

1  1 

II 
II 
II 
II 
II 

12 
jf 

.0653 
.0664 
.0676 
.0688 
.0699 

.1566 
.1586 
.1606 
.1626 
.  1646 

.O7II 

.1423 

-i66g 

.0723 

•0735 
.0748 
.0760 
.0772 

.1443 
.1458 
.1473 
.1489 
.1504 

.1687 
•  i7of 
•1728 
-1749 
.1770 

.0728 

•0739 
.0750 
.0761 
.0772 
•0783 

•0785 

.1519 

.1792 

.0798 
.0811 
.0824 
.0837 
.0850 

•1535 
.1556 
.1566 
.1582 
•i59f 

.1813 
.1835 
.1857 
.1879 

.1001 

3 
4 

1 

9 

i 

tl 

10.8 

'I'5 
16.2 

18.9 

21.6 

24.3 

23 

2-3 

X 

10.4 

13.0 
15.6 

18.2 

20.8 

23.4 

22 

2.2 

75 

IO.O 

»*-5 

15  0 

17-5 

20.  o 

22.5 

21 

2.1 

7.2 
9.6 

T2.0 

M  4 

16.8 
19.2 

21.6 

2O 

2.0 

.079  s 

.0805 
.0818 
.0829 
.0841 
.0853 

.0863 

.1613 

.1923 

.0877 
.0890 
.0904 
.0913 
.0932 

.1629 
.1645 
.1661 
.1677 
•1693 

•  1946 
.1969 
.1992 
.2015 

.2062 

3 

4 
5 
6 

7 
8 
9 

i 

2 

3 
4 

I 

9 

i 

2 

3 

4 
5 

6.9 

9.2 
"•5 
13.8 

16.1 
18.4 
20.7 

19 

3--1 
57 

7  6 
9-5 
11.4 

13-3 
15-2 
17.1 

15 

1-5 
3  -o 

4.5 

6.0 

7-5 

6.6 
8.8 

11.0 

„, 

15-4 
«7.< 

19.8 

18 

1.8 
3-6 
5-4 

7-2 

9.0 
10.8 

is.4 

14-4 
16.2 

14 

::l 

4.2 

5-6 

r 

6-3 

8.4 
,0.5 

12.6 

14.7 
16.8 
18.9 

'7 

1-7 
3-4 

5-i 

6.8 
8-5 

10.2 

11.9 

I3.6 
ISO 

13 

1-3 
2.6 

3.9 

i", 

6.0 
8.0 

IO.O 
12.  0 

14.0 

16  o 
18  o 

16 

1.6 

r* 

i.i 

9.6 

II.  2 
12.8 
M'4 

12 

1:1 

4.8 

6.0 

.0946 

.1709 

.0875 
.0883 

.0906 
.0912 
.0924 

12 
12 
12 
12 
1  2 

.0966 

.0975 
.0989 
.  1004 
.  1019 

.  1726 
.1742 
.1758 
•1775 

.1825 
.1842 

•1859 
.1876 

•1893 

.2086 
.2110 
•  2134 
•2158 
.2183 

.22C7 
.2232 
.2258 
.2283 
.2309 
•2334 

.0949 

.0961 

.0974 

.0985 

.0999 

12 
12 
12 
12 

13 

12 

.1034 

.1049 
.1064 
.1079 
.1094 

.1110 

.IOI2 

.1126 

.1910 

.1927 

.1944 
.1961 

.1979 
.1906 

.2366 

.  1025 

-I03f 
.  1056 
.  1063 
•  !O77 

!i 

13 
3 

\ 

13 
13 
i§ 
14 
H 
13 
14 
14 
U 
H 
14 
H 
14 
U 

.1142 
.1158 

.1174 
.1196 

.1205 

.2387 
.2413 
.2440 
.2467 
.2494 

.1090 

•  1103 
.1116 
.1130 

.1143 
.1157 
1176 
.1184 
.1198 

\122\ 
.1239 

1223 

.2013 

.2521 

.1240 
.1257 
.1274 

.1291 
•  I3°8 

.2031 
.2049 
.2065 
.2084 

.2102 

•2549 

•2576 
.2604 
.2633 
.2661 

I 

9 

10.5 

12.* 
13-5 

] 

I 

9.8 

II  .2 
12.6 

[I  I 

I.I  I 

9.1 
10.4 
11.7 

o  6 

0  0. 

8.4 
9.6 
10.8 

5 

.1325 

2I2O 

.2690 

•1343 
.1361 

•1379 
•1397 
.1415 

.2138 
.2156 

•2174 
.2192 
.2210 
.222§ 

.2719 

.2748 
.2778 
.280? 
•2837 

3 

4 

5 
6 

I 

9 

3«3  3 
4-4  4 

i:*  1 

I'll 

9.99 

0  O.I 

0  0.2 
0  0.2 
00.3 

00.3 
o  0.4 

00.4 

.1268 
.1282 

•1296 
.1311 
•1325 
•1339 

.1433 

.286? 

.1452 
.1476 
.1489 
.1503 

•I52f 

.2247 
.2265 
.2284 
.2302 
.2321 

.2898 
.2923 
.2959 
.2991 
.3022 

•I547 

2339 

•3054 

i  ° 

Yers. 

d. 

Exsec. 

d. 

o    / 

Yers.   d. 

Kxser.   d.            P.  P. 

445 


TABLE    X.— NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS. 

4O°-5O°  5O°-6O° 


'  \  Vers.   d. 

Kxsec. 

d. 

c    , 

Vers.   d. 

Exsec.   d. 

p.  P. 

40  0 

10 
20 
30 
40 
50 

41  0 

10 
20 

30 
40 

50 

•2339 

19 

19 
19 
19 
19 
19 
19 
19 
19 
19 
19 

lil 
19 

20 

19 

2O 

•3054 

32 

3! 

33 
33 
34 
33 
34 
34 

35 

36 
37 
37 

3 

38 
38 
39 
38 

39 
39 
40 
40 
40 

41 
41 
42 

43 
43 

43 
44 
44 
44 
45 
45 

47 
47 
4? 
48 

48 
48 
49 
50 
5? 

If 

52 

53 

53 
53 

50  0 

•3572 

22 
22 
22 
22 
22 

•5557 

53 
54 
54 
55 
56 
56 

5? 
58 
58 
59 
59 
60 

61 
6T 
62 

62 
63 
64 
64 
65 

65 

66 
67 

68 
68 
69 
70 

7o 

72 

73 
73 
74 
75 
75 
77 

78 
79 
80 
81 
82 
82 
83 
84 
85 

86 
88 
88 
89 
96 

94 

93 
94 
95 
97 
98 

99 

100 

987654 

•  2358 
.2377 
•  2396 
.2415 
•  2434 

.3086 

•3151 
.3183 
.3217 

10 
20 
30 
40 
50 

51  0 

10 

20 

30 
40 

50 

•3594 
.3617 

•3639 
.3661 

.3684 

.5611 
.5666 
•  5721 
•  5777 
•5833 

2  1 

4: 

5  t 
6 

7< 

I 

1 

.8 
•  7 

.6 
['S 

•4 

i-3 

7.2 

5.i 

3 

[.6 

2.4 

J.9 

4.0 
4.8 

5.6 

5.4 

7-2 

2 

0.2 

i  -4 

2  .  I 

a.  8 

3-5 

4.2 

4.0 
5-t 
6-3 

I 
!o.i 

I  .2 

1.8 
2.4 
3-6 

4.2 
4.8 
5-4 

9 

o.§ 
1.9 

.0 

z.o 

2.5 
3-o 

3-5 

4.0 

4-5 

8 

0-8 
i  -7 

0. 

i  . 

2. 
2. 

2 

3 

3 

j 

o 

8 

2 

6 
o 

4 

8 

2 

6 

7 

.2453 

•3250 

.3707 

22 
22 

23 

22 

23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 
23 

24 
23 

23 
24 

23 
24 
24 
24 
24 
24 

24 
24 
24 
24 
24 
24 
24 

24 
24 

24 
24 
24 
25 
24 

24 

25 

25 

24 

.5890 

.2472 
.2491 
.2516 
.2529 
.2549 

.3284 
•3352 

•338S 
•  3421 

•3729 
•  3752 
•3775 
•379? 
.3820 

•594? 
.600^ 
.6064 
.6123 
.6182 
.6242 

42  0 

.2568 

•3456 

52  0 

10 
20 

40 
50 

53  0 

•3843 

10 

20 

30 

40 

50 

.2588 
.260? 
.2627 
.2647 
.2666 

•3491 
•352? 
.3563 
•3599 
•3638 

•  3865 
.3889 
.3912 
•3935 
•3958 

•  6303 

•6365 
.6427 
.6489 
.6552 

3 

4 
5 
6 

7 
8 

9 

0.4  C.2 

I  .2 

'•5 
I.fc 

2.1 
2.4 
2.7 

5 

0.1 
1.0 

I  .2 

1  .4 

1.6 

i.S 

0.4 

O.c 

o.c 

0.7 

0.1 

o.c 

4 

4-7 
5-7 

6.g 
7-6 
8-5 

3 

3--1 

4  :- 

1:1 

7-6 
2 

2 
6 

o 
7 
5 

7 
[ 

43  0 

.2686 

20 
2O 
20 
2O 
20 

.3673 

.3982 

.6615 

10 

20 

30 
40 

50 

•  270^ 

.2725 
.2746 

'2786 

.3716 
•3748 
.3786 
.3824 
.3863 

10 

20 
30 
40 

50 
54  0 

10 
20 

30 

40 

50 

.4005 
.4023 
.4052 

•  4075 
•  4098 

.6681 
.6746 
.6811 
.6878 
•6945 

2 

3 
4 

1 

9 
25 

1.3  1  .  i 

0.9  0.7 

o.  ^ 

0 

3 

44  0 

.2806 

20 
20 
20 
20 
20 
20 

.3901 

.4122 

.7013 

1.9 

2.6 

»«t 

2  .1 

i  .§  1.6 

O.  / 

I  .0 

0.4 
0.6 

10 
20 

30 

40 

.2827 
.2847 
.286? 
.2888 
.290^ 

•3941 
.3980 
.4026 
.4066 
.4101 

Ui6§ 

•  4193 
•  4216 

.4246 

.7081 

.7150 
.7226 
.7291 
.7362 

3-9 

4-5 
S-2 

5-8 

2 

3-: 
3-{ 

4  •  •• 

4A 

5 

2-; 

3-5 

3  ' 

24 

2.1 

2.4 

24 

I  .  c  O 

1.7  1 

2.0  T 
2.21 

23 

2-3 

4-7 
7.6 

9.4 

11.7 
14.1 

16.4 

;8.8 

21.  I 

20 

2.6 

4.1 

6.1 

8.2 
10.2 
12-3 

14-3 
76.4 
.8.4 

18 

3'Z 
5-5 

7-4 
y.2 
i  .  i 

4.8 
6.6 

•9 
.6 

.2 

•3 

23 

2  -3 
4.6 

6.9 

9.2 

"•5 

r6.I 
18  4 

20.7 

20 

2.0 
4  O 

6.0 

8.0 

IO.O 
12.0 

I4.0 

16  o 
18  o 

45  0 

.2929 

.4142 

55  0 

10 

20 

30 
40 

50 

56  0 

10 
20 

30 
40 

50 

57  0 

10 

20 

30 

40 

50 

58  0 

10 
20 

30 
40 

50 

59  0 

10 
20 

30 
40 

50 

60  0 

.4264 

•7434 

10 

20 

30 
40 

50 

46  0 

10 

20 

30 
40 

50 

.2949 
.2970 
.2991 
.3011 
.3032 

20 
21 
20 
21 
21 
21 

.4225 
.4267 
•4309 
.4352 

.4286 
•  4312 
•4336 
.4360 
•  4384 

.750? 

.7655 
•  7730 
.7806 

2  5- 
3  7- 

4  10. 
5  12. 
615. 

8  20. 

9  22. 

22 

I   2  . 
2   4- 

3  6. 

4  9- 
5  "• 
6  13. 

ill:. 

g  20. 

1  5 
5  7 

2  10 

7  12 

J  15 

3  »7 

»  20 
)  22 

2: 

i  2 

a  8 

>  ii 
5  13 

7  15 
>  17 
I  19 

l 

.0 

•  5 

.0 

•5 

.  c 

•5 
2 

.2 

•4 

.6 

.8 

.0 

.2 

•  4 

,6 

.8 

I 
t  i 

>-  : 

5  1 

1  - 

,  i 
i 

;: 

?  !' 

4-9 
7  3 

9  £ 
13.1 

'4  7 

7-1 
19.  e 

22  .C 

21 
B.f 

6\ 

B.< 

10.7 

2.0 

15.4 

9 

•9 
•9 
8 

7.8 

)-7 

.7 

•6 
•  <> 

'•Si 

7.2 

9.6 

12  .C 
14.4 

16.8 
19.2 

21.6 

21 

2  1 

4  2 
6.3 

!  8.4 

10.= 
12.6 

14.7 
16.8 
18.  c, 

[9 

I  ,C 

3-8 
5-7 

7.6 
9-5 
i-4 

3-3 

5-2 

7-1 

•3053 

.3074 
.3095 

'.313? 
•3150 

•439^ 

.4408 

•7883 

21 
21 
21 
2l 
21 

•  4439 
.4483 
•  452? 
.4572 
.461? 

•  4432 

•  4456 
.4486 

•4505 
.4529 

.7966 
.8039 
.8118 
.8198 
.8279 

47  0 

.3180 

21 
21 
21 
21 
21 
21 
22 
21 
22 
2l 
22 
22 
-  22 
22 

*)* 

22 

22 

•  4663 

•4553 

8361 

10 

20 

30 

40 

50 

48  0 

10 

20 

30 

40 

50 

49  0 

10 
20 

30 
40 

50 

50  0 

.3201 
.3222 
•3244 
•  3265 
-3287 

•4708 
-4755 
.4802 

.4849 
•  4896 

•4578 
.4602 
.4627 

•  4651 
.4676 

•  8443 
•  8527 
.8611 
•  8697 
.8783 

•3308 

•4945 

.4701 

.8871 

•3330 
•3352 
•3374 
•3395 

•4993 
.5042 
.5091 
.5141 
.5192 

•4725 
•  4750 

•  4775 
.4800 
.4824 

•8959 
•  9°48 
•9r39 
.9230 
.9322 

•3439 

•  5242 

.4849 

25 
25 
25 
25 

25 

25 

.9416 

•  3461 
.3483 
•35o5 
•352? 
•3550 

•5294 
•5345 
•539? 
.5450 
.5503 

•  4874 
•  4899 
•  4924 
•  4949 
.4975 

.9510 
.9606 

.9703 
.9801 

.99°° 

•3572 

•5557 

.5000 

I  OOOO 

Fxseo   d 

Vers   '  d 

Fxseo.  '  d. 

P.  P. 

446 


TABLE    X.— NATURAL  VERSED  SINES  AND  EXTERNAL  SECANTS. 

6O°-7O°  -         7O°-8O° 


1  °  ' 

Vers.   d. 

Kxsec. 

d. 

o  ; 

Vers.  |  d. 

Exsec.  !  d. 

P.  P. 

60  0 

10 
20 

30 
40 

50 

61  0 

10 
20 

30 
40 

50 

62  0 

10 
20 

30 
40 

50 

63  0 

10 

20 

30 
40 

50 

.5000 

25 

25 
25 
25 

% 

25 
2? 

25 
25 
25 
26 

25 

ll 

26 

11 

26 
26 
26 
26 
26 

26 

26 
26 

26 
26 

26 
26 

26 

26 

26 
26 

26 

l.OOOO 

101 
102 
103 
JO? 

106 
107 
109 

IIO 

HI 

H3 
114 
116 

n? 
H8 

120 
121 

123 

125 

126 

128 
130 
131 
133 
135 
137 
139 

146 

143 
144 

146 
148 
I51 

152 

155 

15? 
159 
161 
164 
165 
169 
171 
174 
177 
179 
182 
185 
1  88 
196 

194 
196 

200 
203 
206 
210 
2I3 

216 
220 
224 
22? 
232 

70  0 

10 
20 
30 
40 
50 

71  0 

10 
20 
30 
40 

5° 
72  0 

10 
20 
30 
40 
50 

73  0 

10 
20 
30 

40 
50 

74  0 

10 
20 
30 
40 
50 

75  0 

10 

20 
30 
40 
50 

76  0 

10 
20 
30 
40 
50 

77  0 

10 
20 
30 
40 
50 

78  0 

10 
20 
30 
40 
50 

79  0 

10 
20 
30 
40 

50 
S()  <) 

.6580 
.6607 

.6634 
.6662 
.6689 
.6717 

27 
2? 

27 
2? 
27 
2? 

1.9238 

1.9473; 
I.97I3 
1-995? 
2.0205 

2.0458 

235 
240 

244 
248 

253 

257 

262 
26? 

276 

276 
281 

287 
292 
298 

304 
310 
316 

322 

3^8 
335 

345 
349 
356 
364 
372 
380 

1388 
396 
406 

!  4H 
424 
434 
444 
454 
465 

476 

48§ 
500 
512 
525 
539 
553 

56? 
582 

598 
614 
631 
649 

66? 
685 
707 
728 

749 
772 

796 
821 
84? 
875 
904 
934 

i 

2 

3 
4 

i 

9 

i 

2 

3 

4 

1 
I 

9 

2 

3 

4 
5 
6 

7 
8 
9 

9 

o.  :. 
».J 

3-« 

4-3 

:-4 

'••3 
7  -a 
8.1 

3 

0.3 

o*g 

1.2 

1.5 

I.fc 

2.1 

2.4 

2-7 

6 

o-« 

1:1 

2.t 

8 

0.8 

i.l 

2.4 

3  -a 

4-  - 

4.-: 
5-<5 

<  -4 

7-  a 

2 

0.2 

'-'•4 

o.< 

O.J 

I.O 

I  .2 

l:i 

1.1 

5 

0  -  5 
I  .  I 
!-6 

2.2 

7 

0.7 
1.4 

2.1 
2.8 

3-5 

4-2 

4-9 
5-6 
6-3 

I 

O.I 

0.2 

0-3 

0.4 

0-5 

0.6 

0.7 
0.8 
0.9 

4 

0.4. 

°-2 

"3| 
1.8 

6 

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[  2 
1.1 

1-4 

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3-' 

4--? 
4.: 
5.4 

9 
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•-.' 
--•- 

J.8 

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5-7 

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8.5 

3 

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I  -4 

5 

0.3 

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2  .C 

*>s 

3-0 

3-: 
4.0 
4-5 

8 
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3-4 

4.9 

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6.1 

7-r 

2 

•:•  .  2 
0.5 

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i  .; 

4 

0.4 

0.8 

1.2 

1.6 

2.0 
2.4 

2.8 

3-2 

3.6 

1 

0.7 

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2.2 

3-o 

3.7 
4.5 

5.2 

6.0 
6.7 

i 

O-I 

03 

0.4 

0.6 

.5025 
.5056 
.5076 
.5101 

•5126 

I  .0101 

I  .  0204 

1.030? 
1.0413 
1.0519 

-5152 

1.0625 

.6744 

2? 
2? 
2? 
27 
2? 
28 

2? 
27 
28 

27 

28 
28 

2? 
28 
28 
2? 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
28 
2§ 
28 

28 

28 

28 

28 

28 

28 

28 
28 

28 

28 

28 

28 

26 

2.0715 

•5177 
.5203 

•522§ 

.5254 
.5279 

1.0735 
1.0846 

1.095? 

I  .  1076 
1  .  1  1  84 

.6772 

.6799 
.6827 

•6854 
.6882 

2  097? 
2.1244 

2.I5I§ 
2.1792 

2.2073 

.5305 

1.1300 

.6910 

2  .  2366 

.5331 

.$356 
•5382 
•5408 
•5434 

1.1418 

1.1536 
1.1657 

1-1778 
i  .  1902 

.693? 
.6965 

.6993 
.7026 

•  7048 

2.2653 
2.2951 
2.3255 
2.3565 
2.3881 

;-546o 

1.2027 

•7076 

2.4203 

.5486 
.5512 
•5538 
•5564 
•  5590 

1.2153 
1.2281 
1.2411 
1-2543 

1.2676 

.7104 
.7132 
.7160 
.7187 
.7215 

2.4531 
2.4867 
2.520§ 

2-5558 
2.59i5 
2.6279 
2.6651 
2.703! 
2.7420 

2.7815 

2.8222 

2.8637 

2  .  906  I 
2.9495 

2.9939 
3.0394 
3.0859 

61  0 

10 

20 

30 
40 

50 

65  0 

10 

20 

30 
40 

50 

66  0 

•5616 

1.2811 

.7243 

.5642 
.5668 
•5695 
.5721 

.5600 
-5826 
•5853 
.5879 
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1-2948 
1.3087 
1.3228 
I-337I 
I-35I? 

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.7299 
•732? 

•735? 
.7412 

1.3662 

1.3816 
1.3961 

1.4114 
i  .  4269 

1.4425 

•  7440 
.7468 
.7496 
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.7552 

3  -a 

3-v 

4.5 
5.  a 

2.7 

3-? 
3  -8 

4-4 

2.2 

2-7 

3-1 
3-6 

i-7 

i.  i 

2-4" 

2.8 

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0.7 
0.9 

i  .6 

t  .2 

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3 
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9 

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2 

3 

4 

5 
t 

7 
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4-y 
29 

2  ! 
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B.J 

ti.d 

14-5 

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90.3 

4.0 
28 

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8-. 

II  .4 

14.5 
17- 

19.  c 

3-1  2.2 

28 

,  2.8 

r  5-6 
5  8.4 

II.  2 
I  14.0 

16.8 
M9-6 

1-3 

2? 

2-7 

1:1 

I.O 

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6-5 

9.2 

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25 

2-5 

•5932 

•"6 
26 

27 

26 
26 

27 

26 

27 
27 

26 

27 
27 

27 

27 
27 

27 
27 
27 

2? 

27 

27 

2? 

27 

2? 
2? 

1.4586 

.7581 

3.I33S 

10 

20 

30 
40 

50 

67  0 

10 
20 

30 
40 

50 

68  0 

10 

20 

30 
40 

50 

69  0 

10 
20 

30 

40 

50 

|70  0 

.5959 
.5986 
.6012 
.6039 
.6066 

1-4747 
1.4912 

i  .  5078 
1.5247 

1.5419 

.7609 

.763? 
•  7665 
.7694 

.7722 

3.1824 

3-2324 
3.2835 
3-3362 
3-3901 

.6092 

1-5593 

7750 

3-4454 

.6119 
.6146 
.6173 
.6200 
.6227 
•6254 
.6281 
.6308 

.6335 
.6362 
.6389 

1.57/0 

i  .  5949 
1.6131 
1.6316 
1.6504 

•  7779 
.7807 

.7835 
.7864 
.7892 

3.5021 
3-5604 
3.6202 
3-6815 
3.7448 

2f  .  1 

27 

2-7 

25-6  25-2 

25  26 

2.5  2.6 

1.6694 

.7921 

•^o 

3.809? 

i.688g 
1.7085 
1.7285 
1.7488 
1.769$ 

•7949 
.7978 
.8005 

.8035 
.8063 

^8 

28 
28 
28 
28 

•70 

3.8765 
3-9451 
4-0158 
4.0885 
4-  1636 

S., 

10.  £ 
13  5 

i6.a 
18.9 

Jl.ri 

-4-3 

7-c 

10.  f 

13-5 
>5-9 

18.5 

21.2 

23-£ 

7.8 

10.4  i 
13.0  i 
15.6  i 

18.2  T 
|20.8  1 

23.4  1 

7-6 

0.2 
2.7 
5-3 

7-8 
°-4 
2.9 

.6416 

•6443 
.6476 

.6498 
.6525 
.6552 

1.7904 

.8092 

28 

28 
28 
28 

29 

28 

28 

4-2408 

1.811^ 

1.8334 
1.8554 
1.8778 
1.9005 

.8126 
.8149 

.817? 
.8205 
.8235 
.8263 

4-3205 
4.4025 
4.4874 
4-5749 
4.6653 

4.758? 

i  9238 

Vers.   d. 

Kxsec.   d. 

0      / 

Vers.   d. 

Kxsec.    d. 

P.  P. 

447 


TABLE   X.— NATURAL   VERSED    SINES    AND    EXTERNAL   SECANTS. 

8O°-85°  85°-9O° 


0      / 

Vers. 

d. 

Exsec.     d. 

0      , 

Vers. 

d. 

Exsec. 

d. 

p.  p. 

80  0 

10 
20 

30 
40 

5° 
81  0 

10 
2O 
30 
40 
50 

82  0 

10 

20 

3° 
40 
50 
83  0 

10 
20 

30 
40 

50 

84  0 

10 
20 
30 

40 
50 

85  0 

.8263 

Ito  tototototo  to  tototototo  to  tototototo  to  tototototo  to  tototototo 
vo  vo  vo  vo  vo  vo  OO)  vo  vo  vo  vo  OO)  vo  vo  vo  Oo>vo  Oo>  vo  vo  O0>vo  O0)vo  OO)  vo  OO)OO)vO  OO) 

4-7587 

966 

999 
1035 
1072 
mi 
1152 

1  196 
1242 
1291 
1343 
1398 

H56 
1519 
1585 
1656 

1812 
1893 
1991 
2091 
2198 
2315 
2440 

2576 
2723 
2884 

3059 
3256 
3466 
369? 

85  0 

10 
20 
30 

40 
50 

86  0 

10 
20 
30 
40 
50 

87  0 

10 
20 
30 
40 
50 

88  0 

10 
20 
30 

40 
50 

89  0 

10 
20 

30 
40 

50 

90  0 

.9I2§ 

to  to  to  to  to  to  to  tototototo  to  tototototo  to  to  to  to  to  to  to  tototototo 

vo  vo  vO>vo  vo  vo  vo  vo  VO)VO  vo  vo  VO  VO  vo  VO>vo  VO  VO  VOVOVOVOVO  VO  vovovovovo 

10.4737 

•3946 
.4229 
.4542 
.4892 
.5284 

•5725 
.6223 
.6789 
.7436 
.8186 
.9041 
1.0047 
i  .1230 
1.2634 

'•43  i  9 
1.6365 
1.8884 
2.2032 
2.6039 
3.1247 
3-8192 
4.7741 
6.1383 
8.1846 

29  29  2§ 

i   2.9  2.9  2.  § 
2  5-9  5-8  5-7 
3  8.8  8.7  8.5 

4  ii.  8  ii.  6  11.4 
5  14.7  14.5  14.2 
6  17.7  17.4  17.1 

7  20.  g  20.3  19.9 
8  23.6  23.2  22.8 
9  26.5  26.1  25.5 

.8292 
•8321 

.8349 
.8378 
.8407 

4.8554 
4.9553 
5.0588 
5.1666 

5.2772 

.9157 
.9186 
.9215 
.9244 

.9273 

10.8683 

II  .2912 

11.7455 
12.2347 
12.7631 

.8435 

5.3924 

.9302 

13.3356 

.8464 

.8493 
.8522 

.8550 
.8579 

5.5121 
5.6363 
5.7654 
5.8998 

6.0396 

.9331 
.9366 

•9418 

.944? 

13-9579 
14.6368 
i  5  .  3804 

16.1984 

17.  1026 

.86o§ 

6.1853 

9475 

18.1073 

.8637 
.8666 
.8694 
.8723- 
•8752 

6.3372 

6.4957 

6.6613 
6.8344 

7-0156 

.9505 
•  9534 
.9564 
•9593 
.9622 

19.2303 
20.4937 

21.9256 
23.5621 
25.4505 

.8781 

7-2055 

.9651 

27-6537 

.8810 
.8839 
.8868 
.8897 
.8926 

7.4046 
7.6138 

7.8336 
8.0651 
8.3091 

.9680 
.9709 
.9738 
•976? 
•9796 

30.2576 
33.3823 
37.2015 

41.9757 

48  .  1  1  46 

•8954 

8.5667 

•9825 

56.2987 

.8983 
.9012 
.9041 
.9070 
.9099 

8.8391 
9.1275 

9-4334 

9.7585 

10.  1045 

.9854 
.9883 
.9912 
.9942 
.9971 

67.7573 
84.9456 

1  13  .5930 
170.8883 

342.7752 

.9123 

10.4737 

I  .  OOOO 

00 

0     / 

Vers.  |  d. 

Exsec.     d. 

0     / 

Vers. 

d. 

Exsec.      d. 

448 


ii 


12 


TABLE   XI.— USEFUL   TRIGONOMETRICAL   FORMULAE. 


i  tan  0  j/i  —  cos  20 

sm  0         = 


cosec  0        Vi  _u  tan20  2  4/i  4-  cot 


4-  cos  0          2  tan  i0 

= f-r—  •=  vers  0  cot  i0. 

cot  £0          i  4~  tan  i^ 

I  COt  0  I 


cos  0 


sec  0 


i  —  vers  0  =  sin  a  cot  0  =  V'l  —  sin2  0=2  cos2  \a  —  i 
sin  0  cot  \a  —  i  —  cos2  \a  —  sin2  \a  —  i  —  2  sin2  ^0. 

i       _  sin  0 sec  0      _  i 

cot  0       cos  0       cosec  a        Vcosec2  0—1 

vers  20  cosec  20  =  cot  a  —  2  cot  20  =  sin  0  sec  0 
sin  20 


=  exsec  a  cot    0  =  exsec  20,  cot  20. 


cot  a        = 


cos  20 

cos  0  sin  20  i  4-  cos  20 


tan  0       sin  a       i  —  cos  20  sm 


=  Vcosec2  0  —  i  =  cot  |0  —  cosec  0. 

c-         vers  0        =i  —  cos  0  =  sin  0  tan  \a  =  2  sin2  ^0  =  cos  0  exsec  0. 
5        exsec  0     =  sec  a  —  i  =  tan  a  tan  \a  =  vers  0  sec  0. 


sn  0         vers  0  cos 
sin 


2  cos  1#  sin  0 


./i  +  cos  0          sin  0          sin  a  sin 
cos  |tf      =  y  — ! 


2  sin  £0  vers  0 

tan  0 


tan  £0       =  vers  0  cosec  tf  =  cosec  0  —  cot  tf  = 


sec  0 


i  -f  cos  0  tan  0  i 

cot  10       =  : —  cosec  0  4~  cot  a  = 


10  sin  0  exsec  0       cosec  0  —  cot  0 


vers  \a     =  i  —  v£(i  4~  cos  *)• 


exsec  \a  =  — i. 

4~  cos  0) 


449 


TABLE   XL— USEFUL   TRIGONOMETRICAL    FORMULA. 


13         sm  20       =2  sin  a  cos  a  = 


26 


27 


2  tan  a 


i  -|-  tan2  a 

14        cos  20      =  cos2  0  —  sin2  0=1  —  2  sin2  0=2  cos2  a  —  i 
i  —  tan2  a 


2  tan  0 
tan  20       =  - 


16         cot  20      =  4  cot  0  —  |  tan  0  = 


tan*  a 

cot2  0—i       i  —  tan2  0 


2  cot  0  2  tan 

T7 

tan 
18         exsec  20  = 


cot  a        i  —  tan*  a       i  —  2  sin4  0 

19  sin  (0  ±  £)  =  sin  0  cos  £  ±  cos  a  sin  £. 

20  cos  (0  ±  ^)  =  cos  0  cos  £  =F  sin  a  sin 

21  sin  0  +  sin  b    =  2  sin  4(0  -f-  ^)  cos  4(0  —  b). 

22  sin  0  —  sin  b    —  2  sin  J(0  —  ^)  cos  \(a  -\-  b). 

23  cos  0  -j-  cos  b  =  2  cos  ^(0  +  ^)  cos  ^(0  —  b). 

24  cos  0  —  cos  b  =  —  2  sin  \(a  -j-  £)  sin  ^(0  —  <£). 


Call  the  sides  of  any  triangle  A,  B,  C,  and  the  opposite  angles  0,  b,  and 
c.     Call  ^  =  i(^4  +  J3  +  C). 


1  /          zx        ^  ~  ^         1  /          zx       ^  -  ^ 
tan  4(0  -  b)  =    A    ,     n  tan  4(0  +  A=   -r- — ^  cot 


cos  ^(a  —  b) 


—  A) 

28 


-  C) 


Area    =  f>(*  -  -4)(^  —  ^?)(j  —  C)  = 


30  2  sm  0 


450 


INDEX. 


Abutments  for  trestles,  167. . 

Accuracy  of  earthwork  computations, 
109. 

Accuracy  of  tunnel  surveying,  189. 

Adjustments  of  instruments,  303. 

Advantages  of  tie-plates,  260. 

Allowance  for  shrinkage  of  earth- 
work, 113. 

American  system  of  tunnel  excava- 
tion, 197. 

Angle-bar   (rail- joint) — efficiency,  255. 

ARCH  CULVERTS,  215. 

Area  of  culverts — method  of  compu- 
tation, 204. 

Area  of  culverts — results  based  on 
observation,  206. 

Area  of  the  waterway — culverts,  203. 

A.  S.  C.  E.  standard  rail  section,  245. 

Austrian  system  of  tunnel  excavat'on, 
197. 

Averaging  end  areas — for  volume  of 
earthwork,  79. 

BALLAST,  220. 

Ballast — cost,  224. 

Ballast— methods  of  laying,  223. 

Barometric  elevations,  6. 

Belgian  system  of  tunnel  excavation, 

197. 

BLASTING,  142. 
Blasting — cost,   147. 
Borrow-pits — earthwork,  102. 
Bowls    (ties),  241. 
Box  CULVERTS,  212. 


Bracing — trestles,   166. 
Bracing  (trestles) — design,  184. 
Bridge-joints    (rail),   257. 
Bridge  spirals,  4. 
Broken-stone  ballast,  221. 
Burnettizing — wooden  ties,  234. 

Caps  (trestle) — design,  184. 

Cars  and  horses — use  in  hauling 
earthwork — cost,  134. 

Cars  and  locomotives — use  in  hauling 
earthwork — cost,  136. 

Carts — use  in  hauling  earthwork — 
cost,  130. 

Cattle-guards,  216. 

Cattle-passes,   218. 

Center  of  gravity  of  side-hill  sections — 
earthwork,  107. 

Central  angle — of  a  curve,  21. 

Chemical  composition  of  rails,  251. 

Cinders   (ballast),  221. 

Circular  lead-rails — switches,  278. 

Classification  of  excavatel  material, 
148. 

COMPOUND  CURVES,  37. 

Compound  curves — application  of  tran- 
sition curves,  56. 

Compound  sections — earthwork,  67. 

Computations  (approximate)  from 
profiles — earthwork,  111. 

Computation  of  products — earthwork, 
90. 

COMPUTATION  CP  VOLUME  OF  EARTH- 
WORK, 76. 

451 


452 


INDEX. 


Connecting  cui've  from  a  curved  track 

to  the  inside,  291. 
Connecting  curve  from  a  curved  track 

to  the  outside,  290. 
Connecting    curve    from     a     straight 

track,   290. 

CONSTRUCTION  OF  TUNNELS,  195. 
Contractor's  profit — earthwork,   140. 
Corbels— trestles,   168. 
Cost  of  ballast,  224. 
Cost  of  earthwork,  126. 
Cost  of  framed  timber  trestles,  174. 
Cost  of  metal  cross-ties,  240. 
Cost  of  pile  trestles,  161. 
Cost  of  rails,  254. 
Cost  of  ties,  232. 

Cost  of  treating  wooden  ties,  236. 
Cost  of  tunneling,  201. 
Creosoting — wooden  ties,  233. 
Cross-country  route,  3. 
CROSSINGS,  300. 
Crossing — one  straight  and  one  curved 

track,  301. 

Crossing — two  curved  tracks,  301. 
Crossing — two  straight  tracks,  300. 
Cross-over  between  two  parallel  curved 

tracks — reversed    curve,    296. 
Cross-over  between  two  parallel  curved 

tracks — straight    connecting    curve, 

295. 
Cross-over      between      two      parallel 

straight  tracks,  293. 
Cross-sectioning — field-work,  10. 
Cross-sectioning — for  volume  of  earth- 
work, 73. 
Cross-sectioning    irregular    sections — 

earthwork,  100. 

Cross-section  method  of  obtaining  con- 
tours,  9. 

Cross-sections — ballast,  222. 
Cross-sections  of  tunnels,  190. 
CULVERTS,  202. 
Curvature      correction  —  volume      of 

earthwork,  103. 

Curve  location  by  deflections,  23. 
Curve   location   by   middle    ordinates, 

27. 


Curve    location   by    offsets    from    the 

long  chor-i,  28. 
Curve  location  by  tangential   offsets, 

26. 
Curve  location  by  two  transits,  26. 

Deflections  for  a  spiral,  49. 

Design  of  culverts — elements,  202. 

Design  of  nut-locks,  268. 

Design  of  pile  trestles,  161. 

Design  of  ti3-plates,  261. 

Design- of  track-bolts,  267. 

DESIGN  OF  TUNNELS,  190. 

DESIGN  OF  WOODEN  TRESTLES,  174. 

Dimensions  of  wooden  ties,  229. 

Ditch -s,  69. 

Drains — tunnels,  195. 

Drill-holes,    position    and    direction — 

blasting,  145. 
Drilling — blasting,  144. 
Driving  spikes,  264. 
Durability  of  metal  ties,  238. 
Durability  of  wooden  ties,  228. 

Early  forms  of  rails,  243. 

EARTHWORK — COST,  126. 

EARTHWORK  SURVEYS,  72. 

Eccentricity  of  the  center  of  gravity 
of  an  earthwork  "rcss-section,  104. 

Economics  of  treated  lies,  236. 

Elements  of  a  1°  curve,  22. 

Elements  of  a  simple  curve,  21. 

English  system  of  tunnel  excavation, 
197. 

Enlargement  of  headings — tunnels, 
196. 

Equivalent  level  sections — earthwork, 
85. 

Equivalent  sections — earthwork,  83. 

Existing  track — determination  of  cur- 
vature, 35. 

Expansion  of  rails,  249. 

Exploding  the  charge — blasting,  147. 

Explosive,  amount  required  in  blast- 
ing, 146. 

Explosives — blasting,  142. 

Extent  of  use— metal  ties,  238. 


INDEX. 


453 


Extent  of  use  of  trestles,  153. 
External  distances  for  a  1°  curve,  318. 
External  distance — simple  curve,  21. 

Factors    of   safety — design    of    timber 

trestles,  180. 

Failures  of  rail-joints,  258. 
Fastenings  for  metal  cross-ties,  240. 
Field-work  for  locating  a  spiral,  52. 
Fire  protection  on  trestles,  173. 
Five-level  sections — earthwork,  92. 
FLOOR  SYSTEMS  OF  TRESTLES,  167. 
FORMATION  OF  EMBANKMENTS,  111. 
Forming  embankments — methods,  115. 

FORM  OF  EXCAVATIONS  AND  EMBANK- 
MENTS, 64. 

Forms  of  rail  sections  (standard),  244. 

Formulae  for  required  area  of  culverts, 
205. 

Foundations — trestles,  165. 

Framed  timber  trestles — cost,  174. 

FRAMED  TRESTLES,  162. 

Free  haul— limit,  124. 

French  system  of  tunnel  excavation, 
197. 

Frogs,  272. 

Frog  angles  —  trigonometrical  func- 
tions, 321. 

Frog  number,  273. 

German  system  of  tunnel  excavation, 

197. 
Grade    line — change,    based    on    mass 

diagram,  123. 
Grade  of  tunnels,  192. 
Gravel  (ballast),  221. 
Ground-levers,  276. 
Guard-rails — switches,  277. 
Guard-rails— trestles,  169. 

Hauling   earthwork — cost,    130. 

Haul  of  earthwork  —  computations, 
116. 

Haul  of  earthwork — method  depend- 
ent on  distance,  137. 

Haul  of  earthwork — profitable  limit, 
140. 

Headings — tunnels,  195. 


I-beam  bridges,  219. 

Instrumental  work  of  locating  curves, 

24. 

Iron-pipe  culverts,  209. 
Irregular  prismoid — volume,  94. 
Irregular  sections — earthwork,  93. 

Joints  of  framed  trestles,  162. 
Kyanizing — wooden  ties,  234. 

Lateral  bracing — trestles,  167. 

Length  of  a  simple  curve,  20. 

Length  of  rails,  248. 

Level — adjustments,  309. 
|  Level  sections — earthwork,  81. 
I  Limitations  in  location,  34. 

Lining  of  tunnels,  193. 

Loading— design    of    timber    trestles, 
179. 

Loading  earthwork — cost,  128. 

Location  surveys,  13. 

Logarithmic    sines    and    tangents    of 
small  angles — table  of,  345. 

Logarithmic    sines,    cosines,    tangents, 
and  cotangents — table  of,  348. 

Logarithmic  versed  sines  and  external 
secants — table  of,  393. 

Logarithms  of  numbers — talle  of.  325. 

Long  chords  for  a  1°  curve,  318. 

Long  chord — simple  curve,  21. 
|  Longitudinal  bracing — trestles,   166. 
j  Longitudinals,  241. 

Loosening  earthwork — cost,   127. 

MATHEMATICAL  DESIGN  OF  SWITCHES, 
278. 

Mass  curve — area,  121. 

Mass  curve — properties,  118. 

Mass  diagram,  117. 
!  Mass  diagram — value,  122. 
|  Metal  cross-ties — cost,  240. 
i  Metal  cross-ties — fastenings.  240. 

METAL  TIES,  238. 

Metal  ties — form  and  dimensions,  239. 

Middle    areas — for    volume    of    earth- 
work, 79. 


454 


INDEX. 


Middle  ordinate — simple  curve,  21. 
Modifications    of    location — compound 

curves,  40. 
Modifications      of      location  —  simple 

curves,   31. 
Mountain  route,  3. 
"Mud"  ballast,  220. 
Mud-sills — trestle  foundations,  166. 
Multiform  compound  curves,  47. 
Multiple-story     construction — trestles, 

163. 

Natural  sines,  cosines,  tangents,  and 
cotangents — table  of,  439. 

Natural  versed  sines  and  external  se- 
cants— table  of,  444. 

Notes — location  surveys,  16. 

Number  of  a  frog— to  find,  273. 

NUT-LOCKS,  266. 

Obstacles  to  location,  29. 
Obstructed  curve — curve  location,  31. 
Old-rail  culverts,  213. 
Open  cuts  vs.  tunnels,  200. 
Ordinates  of  a  spiral,  48. 

"Paper  location,"  13. 

Pile  bents,   155. 

Pile-driving  formulae,  159. 

Pile-driving  methods,  157. 

Pile  foundations  for  trestles,  165. 

Pile-points  and  pile-shoes,  160. 

PILE  TRESTLES,  155. 

Pile  trestles — cost,  161. 

PIPE  CULVERTS,  208. 

Pipe  culverts — construction,  208. 

Pit  cattle-guards,  216. 

Ploughs — use  in  loosening  earth,  127. 

Point  of  curve,  21. 

Point  of  curve  inaccessible — curve  lo- 
cation, 33. 

Point  of  tangency,  21. 

Point  of  tangency  inaccessible — curve 
location,  30. 

Point-rails  of  switches — construction, 
275. 


Point-switches,  275. 

Portals   (tunnel) — excavation,  199^ 

Posts  (trestle)— design,  1£2. 

PRELIMINARY  SURVEYS,  8. 

Preservation  of  ties — cost,  236. 

PRESERVATIVE  PROCESSES  FOR  wooi> 
EN  TIES,  232. 

Prismoidal  correction  (approximate) 
for  irregular  prismoids,  99. 

Prismoidal  correction  (true)  for  ir- 
regular prismoids,  95. 

Prismoids,  72. 

Radii  of  curves — table,  314. 

RAILS,  243. 

Rail  expansion,  249. 

Rail-gap  at  joints — effect,  256. 

RAIL-JOINTS,  255. 

Rails — chemical  composition,  251. 

Rails— cost,  254. 

Rail  testing,  252. 

Rail  wear  on  curves,  253. 

Rail  wear  on  tangents,  252. 

RECONNOISSANCE  SURVEYS,  1. 

Renewals  of  ties — regulations,  231. 

Repairs,  etc.,  of  plant  for  earthwork — 

cost,  139. 
Replacement  of  a  compound  curve  by 

a  curve  with  spirals,  58. 
Replacement  of  a  simple  curve  by  a 

curve  with  spirals,  53. 
Requirements  for  a  perfect  rail-joint,, 

255. 

Requirements — spikes,  263. 
Requirements — track-bolts,  266. 
Roadbed— width,  67. 
Roadways  for  earthwork — cost,  138.. 
Rock  ballast,  221. 
Rules  for  switch-laying,  298. 
Ruling  grade,  2. 

Scrapers — use  in  earthwork — cost,  133. 
Screws     and     bolts     (rail-fastenings),. 

264. 

Setting  tie-plates—methods,  262, 
Shafts— tunnels,  193. 
Shaft   (tunnel) — surveying,  187., 


INDEX. 


455 


Shells  and  small  coal — ballast,  221. 

Shoveling  (hand)  of  earthwork — cost, 
128. 

Shrinkage  of  earthwork,  111. 

Side-hill  work— earthwork,  100. 

Sills  (trestle)— design,  184.      . 

SIMPLE  CURVES,  18. 

Slag  (ballast),  221. 

Slide-rule — for  earthwork  computa- 
tions, 90. 

Slopes — earthwork,  65. 

Slope-stakes — position,  75. 

Sodding — effect  on  slopes,  70. 

Spacing  of  ties,  229. 

Span— trestles,  164. 

Specifications  for  earthwork,  148. 

Specifications  for  wooden  ties,  230. 

SPIKES,  263. 

Spikes— driving,  264. 

Spirals — required   length,  48. 

Spreading  earthwork — cost,  138. 

Stadia  method  of  obtaining  topog- 
raphy, 12. 

Standard  angle-bars,  259. 

Standard  stringer  bridges,  219. 

Steam-shoveling — earthwork,  129. 

Stiffness  of  rails — effect  on  traction, 
247. 

Stone  box  culverts,  212. 

Stone  foundations  for  framed  trestles, 
166. 

Straight  connecting  curve  from  a 
curved  main  track,  292. 

Straight  frog-rails — effect,  280. 

Straight  point-rails — effect,   281. 

Strength  of  timber,  176. 

Strength,  required  elements — trestles, 
175. 

Stringers  for  trestles — design,  180. 

Stringers — trestles,   167. 

Stub  switches,  273. 

Subchord— length,  19. 

Subgrade — form,  68. 

Superelevation  of  the  outer  rail  on 
curves — general  principles,  43. 

Superelevation  of  the  outer  rail  on 
curves  on  trestles,  170. 


Superelevation  of  ths  outer  rail  on 
curves — practical  rules,  45. 

Superintendence  of  earthwork — cost, 
139. 

Supported  joints,  257. 

Surface  cattle-guards,  217. 

Surface  surveys — tunneling,  185. 

SURVEYING — TUNNELS,  185. 

Suspended  joints,  257. 

Switchbacks,  4. 

SWITCH  CONSTRUCTION,  271. 

Switch-laying — practical  rules,  298. 

Switch  leads  and  distances,  321. 

Switch-stands,  276. 


Tamping — blasting,  146. 

Tangent  distance — simple  curve,  21. 

Tangents  for  a   1°  curve,  318. 

Temperature  allowances — rails,  250. 

Terminal  pyramids  and  wedges — 
earthwork,  65. 

Testing  rails,  252. 

Three-level    sections — earthwork,    87. 

"  Throw  "  of  a  switch,  279. 

TIE-PLATES,  260. 

Tie-rods,  276. 

TIES,  226. 

Ties— cost,  232. 

Ties  on  trestles,  170. 

Tile  pipe  culverts,  211. 

Timber  for  framed  trestles,  173. 

Timber  for  pile  trestles,  157. 

Timber,  strength,  176. 

Topographical  maps,  use  of,  5. 

TRACK-BOLTS,  266. 

Transit — adjustments,  304. 

TRANSITION  CURVES,  43. 

Transition  curves — fundamental  prin- 
ciple, 43. 

Transition  curves — tables  of,  322. 

TRESTLES,  153. 

TRESTLES — FRAMED,  162. 

TRESTLES — PILE,  155. 

Trestles — posts — design,  182. 

Trestles  —  required  elements  of 
strength,  175. 


456 


INDEX. 


Trestles— sills— design,  184. 

Trestles — stringers — design,   180. 

Trestles  vs.  embankments,  154. 

TUNNELS,  185. 

Tunneling — cost,  201. 

Tunnel    spirals,    5. 

Turnout     (double)     from    a    straight 

track,  287. 
Turnout    from    the    inner    side    of    a 

curved   track — dimensions,   286. 
Turnout    from    the    outer    side    of    a 

curved  track — dimensions,  284. 
Turnouts  with  straight  point-rails  and 

straight  frog-rails — table  of,  321. 
Two-level     ground  —  for     volume     of 

earthwork,  80. 
Two  turnouts  on  the  same  side,  289. 

Underground  surveys,   188. 
Unit  chord — simple  curves,  19. 
Upright   switch-stands,  276. 
Useful  trigonometrical  formulae — table 
of,  449. 


Valley  .route,  2. 

Ventilation  (tunnel)  during  construc- 
tion, 199. 

Vertex  inaccessible — curve  location, 
30. 

Vertex — of  a  curve,  21. 

VERTICAL  CURVES,  61. 

Vertical  curves — form   of  curves,  62. 

Vertical  curves — requ'rcd  length,  Cl. 

Vulcanizing — wooden  ties,  232. 


Waterway  required  for  culverts,  203. 

Wear  of  rails  on  curves,  253. 

Wear  of  rails  on  tangents,  252. 

Weight  of  rails,  246. 

Wellhouse  process  —  for  preserving 
wooden  ties,  235. 

Wheelbarrows — use  in  hauling  earth- 
work— cost,  132. 

Wooden  box  culverts,  212. 

"Wooden"  spikes,  266. 

Wooden  ties,  227. 


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